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FD(2)8620 20018 y FG(a)p FL(\()p FG(k)45 b FL(\))368 b(=)12881 18672 y Fz(n)13891 19455 y FD(1)p FF(;)393 b Fy(if)339 b FF(k)364 b Fy(o)-49 b(dd)13891 20418 y FD(0)p FF(;)345 b Fy(if)339 b FF(k)363 b Fy(even)21151 20018 y FJ(with)465 b FG(a)p FL(\(1\))369 b(=)f FH(4)p FG(a)p FL(\(1\))h(=)g(1)p FJ(.)p Black 4602 22498 a(\(c\))p Black 651 w FH(4)p FG(a)p FL(\()p FG(k)340 b FL(+)295 b(1\))g(+)g FH(4)p FG(a)p FL(\()p FG(k)45 b FL(\))368 b(=)h FG(k)340 b FL(+)295 b(2)1793 b FJ(with)465 b FG(a)p FL(\(1\))369 b(=)f FH(4)p FG(a)p FL(\(1\))h(=)f(1)p FJ(.)p Black 4536 24977 a(\(d\))p Black 651 w FH(4)p FG(a)p FL(\()p FG(k)340 b FL(+)295 b(2\))369 b(=)f FH(4)p FG(a)p FL(\()p FG(k)45 b FL(\))295 b(+)g(1)1793 b FJ(with)465 b FG(a)p FL(\(1\))369 b(=)f FH(4)p FG(a)p FL(\(1\))h(=)f(1)p FG(;)221 b FH(4)p FG(a)p FL(\(2\))370 b(=)e(2)p FJ(.)p Black 4602 27457 a(\(e\))p Black 651 w FH(4)8094 26975 y FD(2)8620 27457 y FG(a)p FL(\()p FG(k)340 b FL(+)294 b(1\))i(+)f FH(4)16071 26975 y FD(2)16596 27457 y 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FG(x)22703 37171 y FD(2)23229 37653 y FL(\))23868 36900 y FG(dF)p 23868 37213 1699 77 v 24010 38312 a(dx)25699 37653 y FL(\()p FG(x)p FL(\))368 b(=)h(2\(1)296 b(+)e(2)p FG(x)p FL(\))p FG(F)181 b FL(\()p FG(x)p FL(\))465 b FJ(with)g FG(F)181 b FL(\(0\))368 b(=)h(1)p FJ(.)4927 40520 y(The)413 b(c)-66 b(o)g(e\261cient)411 b(of)i FG(x)15790 40038 y FF(n)16831 40520 y FJ(in)f(the)i(p)-66 b(ower)414 b(series)f(exp)-66 b(ansion)412 b(of)i(the)f(solution)h FG(F)181 b FL(\()p FG(x)p FL(\))413 b FJ(of)h Fw(any)g FJ(of)f(the)6913 42446 y(fol)66 b(lowing)465 b(di\256er)-66 b(ential)463 b(e)-66 b(quations:)p Black 4602 45044 a(\(b\))p Black 651 w FL(\(1)295 b FH(\241)h FG(x)10432 44562 y FD(2)10958 45044 y FL(\))11597 44291 y FG(dF)p 11597 44604 V 11739 45704 a(dx)13428 45044 y FL(\()p FG(x)p FL(\))368 b(=)h(\(4)295 b(+)g(3)p FG(x)h FH(\241)f FL(2)p FG(x)24088 44562 y FD(2)24910 45044 y FL(+)f FG(x)26955 44562 y FD(3)27481 45044 y FL(\))p FG(F)181 b FL(\()p FG(x)p FL(\))295 b(+)g(1)465 b FJ(with)g FG(F)181 b FL(\(0\))368 b(=)h(0)p FJ(.)p Black 4602 48316 a(\(c\))p Black 651 w FL(\(1)239 b FH(\241)f FG(x)10318 47834 y FD(2)10844 48316 y FL(\))11483 47563 y FG(d)12159 47081 y Fv(2)12767 47563 y FG(F)p 11483 47876 2308 77 v 11625 49050 a(dx)13040 48666 y Fv(2)13923 48316 y FL(\()p FG(x)p FL(\))368 b(=)h(\(4)238 b(+)g(5)p FG(x)h FH(\241)f FL(2)p FG(x)24355 47834 y FD(2)25120 48316 y FL(+)f FG(x)27108 47834 y FD(3)27634 48316 y FL(\))28273 47563 y FG(dF)p 28273 47876 1699 77 v 28415 48976 a(dx)30104 48316 y FL(\()p FG(x)p FL(\))h(+)g(\(3)g FH(\241)h FL(4)p FG(x)f FL(+)g(3)p FG(x)40275 47834 y FD(2)40801 48316 y FL(\))p FG(F)181 b FL(\()p FG(x)p FL(\))439 b FJ(with)g FG(F)181 b FL(\(0\))369 b(=)f(0)p FJ(,)7046 50083 y FG(dF)p 7046 50396 V 7188 51496 a(dx)8877 50836 y FL(\(0\))h(=)g(1)p FJ(.)4983 53703 y(The)471 b(c)-66 b(o)g(e\261cient)467 b(of)j FG(x)16017 53221 y FF(n)p FD(+1)18316 53703 y FJ(in)f(the)h(p)-66 b(ower)470 b(series)g(exp)-66 b(ansion)469 b(of)h(the)g(solution)h FG(F)181 b FL(\()p FG(x)p FL(\))469 b FJ(of)h Fw(any)h FJ(of)6913 55630 y(the)465 b(fol)66 b(lowing)464 b(di\256er)-66 b(ential)463 b(e)-66 b(quations:)p Black 4536 58228 a(\(d\))p Black 651 w FL(\(1)295 b FH(\241)h FG(x)10432 57746 y FD(2)10958 58228 y FL(\))11597 57474 y FG(dF)p 11597 57788 V 11739 58887 a(dx)13428 58228 y FL(\()p FG(x)p FL(\))368 b(=)h(\(6)295 b(+)g(2)p FG(x)h FH(\241)f FL(4)p FG(x)24088 57746 y FD(2)24910 58228 y FL(+)f(2)p FG(x)27605 57746 y FD(3)28132 58228 y FL(\))p FG(F)181 b FL(\()p FG(x)p FL(\))294 b(+)h(2)p FG(x)465 b FJ(with)g FG(F)181 b FL(\(0\))369 b(=)f(0)p FJ(.)p Black 4602 61500 a(\(e\))p Black 651 w FG(x)p FL(\(1)296 b FH(\241)f FG(x)11171 61018 y FD(2)11697 61500 y FL(\))12336 60747 y FG(d)13012 60265 y Fv(2)13620 60747 y FG(F)p 12336 61060 2308 77 v 12478 62234 a(dx)13893 61850 y Fv(2)14776 61500 y FL(\()p FG(x)p FL(\))368 b(=)h(\(1)295 b(+)g(6)p FG(x)h FL(+)f(3)p FG(x)25415 61018 y FD(2)26236 61500 y FH(\241)g FL(4)p FG(x)28953 61018 y FD(3)29775 61500 y FL(+)g(2)p FG(x)32471 61018 y FD(4)32997 61500 y FL(\))33636 60747 y FG(dF)p 33636 61060 1699 77 v 33778 62160 a(dx)35467 61500 y FL(\()p FG(x)p FL(\))g(+)g(\()p FH(\241)p FL(6)g FH(\241)h FL(4)p FG(x)44022 61018 y FD(2)44843 61500 y FL(+)f(4)p FG(x)47539 61018 y FD(3)48065 61500 y FL(\))p FG(F)181 b FL(\()p FG(x)p FL(\))6913 64293 y FJ(with)465 b FG(F)181 b FL(\(0\))369 b(=)f(0)465 b FJ(and)17942 63540 y FG(d)18618 63058 y Fv(2)19227 63540 y FG(F)p 17942 63853 2308 77 v 18084 65027 a(d)18760 64643 y Fv(2)19369 65027 y FG(x)20382 64293 y FL(\(0\))369 b(=)g(2)p FJ(.)597 b FL(\()p FJ(or)27756 63540 y FG(dF)p 27756 63853 1699 77 v 27898 64952 a(dx)29587 64293 y FL(\()p FH(\241)p FL(2\))370 b(=)34165 63770 y FB(\241)p FD(4)p 34165 63987 1203 54 v 34296 64751 a(27)35500 64293 y FJ(,)36496 63540 y FG(dF)p 36496 63853 1699 77 v 36638 64952 a(dx)38328 64293 y FL(\(2\))f(=)41872 63770 y FD(28)p 41872 63987 941 54 v 42107 64751 a(9)42946 64293 y FL(\))p Black 1675 67326 a FJ(11.)p Black 1600 w FL(Max)4052 68212 y FF(k)24 b FB(2f)p FD(1)p FF(;:::)r(;n)p FB(g)8703 67326 y FG(k)340 b FH(\242)295 b FL(\()p FG(n)h FH(\241)f FG(k)340 b FL(+)295 b(1\))p FJ(.)p Black 26475 74617 a FL(3)p Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black Black Black 1675 1424 a FJ(12.)p Black 2100 w FL(Max)4052 2310 y Fu(A)p FB(2)p Fy(Part)o FD(\(1)p FF(::n)p FD(\))9703 1424 y FH(j)p Ft(A)p FH(j)365 b(\242)g FL(Max)12762 2260 y FF(A)p FB(2)p Fu(A)15243 1424 y FH(j)p FG(A)p FH(j)1119 b FJ(wher)-66 b(e)559 b(Part)p FL(\(1)p FG(::n)p FL(\))h FJ(is)g(the)f(c)-66 b(ol)66 b(le)-66 b(ction)557 b(of)i(set)g(p)-66 b(artitions)558 b(of)h(the)g(set)4052 3771 y FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d (n)p FH(g)p FJ(,)516 b FH(j)p Ft(A)p FH(j)505 b FJ(is)g(the)f(numb)-66 b(er)504 b(of)h(blo)-66 b(cks,)514 b(and)505 b FL(Max)29704 4606 y FF(A)p FB(2)p Fu(A)32186 3771 y FH(j)p FG(A)p FH(j)g FJ(is)g(the)f(size)h(of)g(the)g(lar)-66 b(gest)504 b(blo)-66 b(ck)504 b(of)4052 5824 y(p)-66 b(artition)463 b Ft(A)p FJ(.)p Black 1675 8857 a(13.)p Black 1712 w FL(Max)4052 9743 y FF(\256)p FB(2)p FD(p)26 b(erm\()p FF(n)p FD(\))8925 8857 y FG(i)p FL(\()p FG(\256)8 b FL(\))107 b FH(\242)g FG(d)p FL(\()p FG(\256)8 b FL(\))1710 b FJ(wher)-66 b(e)380 b FL(p)36 b(erm\()p FG(n)p FL(\))380 b FJ(is)g(the)h(set)f(of)g (p)-66 b(ermutations)379 b(of)h FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)k(;)c(n)p FH(g)p FJ(,)398 b FG(i)p FL(\()p FG(\256)8 b FL(\))380 b FJ(is)g(the)4052 11204 y(length)355 b(of)h(the)f(longest)h(incr)-66 b(e)g(asing)353 b(subse)-66 b(quenc)g(e)355 b(and)h FG(d)p FL(\()p FG(\256)8 b FL(\))356 b FJ(the)g(longest)g(de)-66 b(cr)g(e)g(asing)353 b(subse)-66 b(quenc)g(e)4052 13130 y(of)464 b(p)-66 b(ermutation)464 b FG(\256)8 b FJ(.)598 b(Se)-66 b(e)563 b FL([)p 0 1 0 0 TeXcolorcmyk(30)p (#cite.s61cjm) [[229 599 241 611] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p FJ(.)p Black 1675 16163 a(14.)p Black 898 w FL(Max)4052 17049 y FF(p)p FB(2)p FF(S)50 b FD(\()p FF(n)p FD(\))7299 16163 y FL(max\()p FG(p)p FL(\))364 b FH(\242)g FL(len\()p FG(p)p FL(\))1885 b FJ(wher)-66 b(e)558 b FG(S)77 b FL(\()p FG(n)p FL(\))558 b FJ(is)g(the)f(set)h(of)f (c)-66 b(omp)g(ositions)556 b(or)i(p)-66 b(artitions)557 b(of)g FG(n)i FL(\()p FJ(the)4052 18510 y(se)-66 b(quenc)g(es,)398 b(with)384 b(or)g(without)g(r)-66 b(e)g(gar)g(d)382 b(to)i(or)-66 b(der,)399 b(of)384 b(p)-66 b(ositive)382 b(inte)-66 b(gers)382 b(which)i(sum)g(to)g FG(n)p FL(\))p FJ(,)400 b FL(max)q(\()p FG(p)p FL(\))4052 20436 y FJ(is)549 b(the)g(size)h(of)f (the)g(lar)-66 b(gest)549 b(p)-66 b(art,)570 b(and)550 b FL(len\()p FG(p)p FL(\))e FJ(is)i(the)f(numb)-66 b(er)548 b(of)i(p)-66 b(arts)549 b(of)g FG(p)p FJ(.)851 b(Se)-66 b(e)549 b(chapter)g(6)4052 22362 y(of)745 b FL([)p 0 1 0 0 TeXcolorcmyk(29)p (#cite.r78) [[128 516 140 528] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p FJ(.)p Black 1675 25395 a(15.)p Black 1838 w FL(Max)4052 26281 y FF(P)131 b FB(2)p FD(ppart)o(\()p FF(n)p FD(\))9179 25395 y FL(#ro)-36 b(ws\()p FG(P)181 b FL(\))j FH(\242)g FL(#cols)q(\()p FG(P)d FL(\))1743 b FJ(wher)-66 b(e)415 b FL(ppart)o(\()p FG(n)p FL(\))g FJ(is)g(the)g(set)g(of)g(plane)f(p)-66 b(artitions)414 b(or)h(Y)-100 b(oung)4052 27742 y(table)-66 b(aux)464 b(of)h FG(n)p FJ(.)597 b(Se)-66 b(e)563 b FL([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.c94) [[208 467 214 479] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)434 b(p.217])p FJ(,)465 b FL([)p 0 1 0 0 TeXcolorcmyk(35)p (#cite.sw86) [[263 467 274 479] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)434 b(p.81])p FJ(,)466 b FL([)p 0 1 0 0 TeXcolorcmyk(17)p (#cite.f97) [[317 467 329 479] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])g FJ(and)602 b FL([)p 0 1 0 0 TeXcolorcmyk(30)p (#cite.s61cjm) [[364 467 375 479] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p FJ(.)p Black 1675 30775 a(16.)p Black 1894 w FL(Max)4052 31661 y FF(G)p FB(2)p FD(graph\()p FF(n)p FD(\))9289 30775 y FG(\302)p FL(\()p FG(G)p FL(\))241 b FH(\242)f FG(\302)p FL(\()p 14313 29708 1026 54 v FG(G)p FL(\))1768 b FJ(wher)-66 b(e)440 b FL(graph\()p FG(n)p FL(\))g FJ(is)h(the)f(set)g(of)g(simple)g(gr)-66 b(aphs)441 b(on)f FG(n)h FJ(vertic)-66 b(es,)443 b FG(\302)p FL(\()p FG(G)p FL(\))4052 33299 y FJ(is)464 b(the)h(chr)-66 b(omatic)464 b(numb)-66 b(er)463 b(and)p 20789 32232 V 465 w FG(G)i FJ(the)g(c)-66 b(omplement)463 b(of)h(gr)-66 b(aph)465 b FG(G)p FJ(.)p Black 1675 36332 a(17.)p Black 1894 w FL(Max)4052 37217 y FF(G)p FB(2)p FD(graph\()p FF(n)p FD(\))9289 36332 y FG(!)48 b FL(\()p FG(G)p FL(\))34 b FH(\242)p 12622 35601 858 54 v 34 w FG(!)48 b FL(\()p FG(G)p FL(\))1675 b FJ(wher)-66 b(e)347 b FL(graph\()p FG(n)p FL(\))h FJ(is)f(the)g(set)g(of)h(simple)f(gr)-66 b(aphs)347 b(on)h FG(n)g FJ(vertic)-66 b(es,)p 48524 35601 V 368 w FG(!)48 b FL(\()p FG(G)p FL(\))368 b(=)4052 38856 y FG(!)48 b FL(\()p 5416 37789 1026 54 v FG(G)o FL(\))465 b FJ(is)g(the)f(indep)-66 b(endenc)g(e)462 b(numb)-66 b(er)464 b(and)h FG(!)48 b FL(\()p FG(G)p FL(\))464 b FJ(is)g(the)h(clique)f(numb)-66 b(er)464 b(of)g(gr)-66 b(aph)465 b FG(G)p FJ(.)p Black 1675 41889 a(18.)p Black 1894 w FL(Max)4052 42774 y FF(G)p FB(2)p FD(graph\()p FF(n)p FD(\))9068 41889 y FL(\(1)224 b(+)g(\242\()p FG(G)p FL(\)\))g FH(\242)g FG(\260)72 b FL(\()p FG(G)p FL(\))1328 b FJ(wher)-66 b(e)432 b FL(\242\()p FG(G)p FL(\))h FJ(is)f(the)h(size)g(of)g(the)f(lar)-66 b(gest)433 b(de)-66 b(gr)g(e)g(e)431 b(of)h(the)h(vertic)-66 b(es)4052 44236 y(and)478 b FG(\260)72 b FL(\()p FG(G)p FL(\))479 b FJ(is)f(the)g(domination)f(numb)-66 b(er)478 b(of)g(the)h(simple)f (gr)-66 b(aph)478 b FG(G)p FJ(.)639 b FL(\()p FG(\260)550 b FJ(is)479 b(the)f(smal)66 b(lest)480 b(size)e(set)4052 46162 y(of)464 b(vertic)-66 b(es)463 b(of)i FG(G)p FJ(,)f(such)i(that)f (every)f(vertex)g(is)g(in)g(the)h(set)g(or)g(adjac)-66 b(ent)463 b(to)i(it.)p FL(\))p Black 1675 49195 a FJ(19.)p Black 650 w FL(Max)4107 50021 y FF(u)p FB(2)p FD(\255)5959 50132 y FA(n)6802 49195 y FG(f)142 b FL(\()p FG(u)p FL(\))112 b FH(\242)g FG(g)48 b FL(\()p FG(u)p FL(\))1709 b FJ(wher)-66 b(e)382 b FH(h)p FL(\255)19105 49394 y FF(k)19673 49195 y FH(i)20190 48713 y FB(1)20190 49540 y FF(k)24 b FD(=1)22343 49195 y FJ(is)383 b(a)f(se)-66 b(quenc)g(e)381 b(of)h(\257nite)e(sets)j (and)f(for)g(e)-66 b(ach)382 b(p)-66 b(ositive)380 b(inte)-66 b(ger)4052 51372 y FG(k)45 b FJ(,)480 b(ther)-66 b(e)476 b(ar)-66 b(e)477 b(functions)f FG(f)619 b FJ(and)478 b FG(g)524 b FJ(fr)-66 b(om)477 b FL(\255)25749 51571 y FF(k)26795 51372 y FJ(to)g FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)k(;)c (k)45 b FH(g)478 b FJ(such)g(that)f(for)g(al)66 b(l)478 b FG(u)393 b FH(2)e FL(\255)47405 51571 y FF(k)47974 51372 y FG(;)614 b(f)142 b FL(\()p FG(u)p FL(\))304 b(+)4052 53298 y FG(g)48 b FL(\()p FG(u)p FL(\))368 b FH(\267)h FG(k)126 b FL(+)82 b(1)p FJ(,)389 b(and)369 b(ther)-66 b(e)368 b(exist)g FG(w)404 b FH(2)369 b FL(\255)23603 53497 y FF(k)24172 53298 y FJ(,)387 b(such)370 b(that)f FG(f)142 b FL(\()p FG(w)36 b FL(\))82 b(+)g FG(g)48 b FL(\()p FG(w)36 b FL(\))366 b(=)j FG(k)126 b FL(+)82 b(1)369 b FJ(and)g FH(j)p FG(f)142 b FL(\()p FG(w)36 b FL(\))82 b FH(\241)g FG(g)48 b FL(\()p FG(w)36 b FL(\))p FH(j)367 b(\267)4052 55224 y FL(1)p FJ(.)4052 57704 y(Note)476 b(that)g(this)h(is)g(a)g(gener)-66 b(alization)474 b(of)j(the)g(ab)-66 b(ove)476 b(items)p 0 1 0 0 TeXcolorcmyk 476 w(11)p (#Item.28) [[378 198 389 210] [1 1 1 [3 3]] [0 0 1]] pdfm Black 477 w(to)p 0 1 0 0 TeXcolorcmyk 477 w(18)p (#Item.35) [[408 198 420 210] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)j(which)e(ar)-66 b(e)477 b(sp)-66 b(e)g(cial)476 b(c)-66 b(ases;)4052 59630 y(se)g(e)464 b(se)-66 b(ction)p 0 1 0 0 TeXcolorcmyk 563 w FL(2)p (#section.2) [[168 180 173 192] [1 1 1 [3 3]] [0 0 1]] pdfm Black 465 w FJ(b)g(elow.)p Black 1675 62663 a(20.)p Black 650 w(The)464 b(numb)-66 b(er)464 b(of)h(gr)-66 b(aphs)465 b(with)g(multiple)f(e)-66 b(dges)464 b(and)h(lo)-66 b(ops)466 b(on)e(two)i(vertic)-66 b(es)463 b(and)i FG(n)295 b FH(\241)h FL(1)465 b FJ(e)-66 b(dges.)p Black 1675 65696 a(21.)p Black 650 w(The)473 b(numb)-66 b(er)472 b(of)g(c)-66 b(onne)g(cte)g(d)471 b(bip)-66 b(artite)471 b(gr)-66 b(aphs)473 b(with)g(p)-66 b(art)473 b(sizes)g FG(n)h FJ(and)f FL(2)p FJ(.)622 b(Se)-66 b(e)p 0 1 0 0 TeXcolorcmyk 572 w FL(Gordon)441 b(Ro)-36 b(yle)p [[472 126 544 138] [1 1 1 [3 3]] [0 0 1]] (http://www.cs.uwa.edu.au/~gordon/) pdfm Black FJ(,)4052 67622 y FI(/www.cs.uwa.edu.au/~gordon/)p Black 1675 70655 a FJ(22.)p Black 650 w(The)464 b(numb)-66 b(er)464 b(of)h FL(\()p FJ(nonc)-66 b(ongruent)p FL(\))462 b FJ(inte)-66 b(ger-side)g(d)462 b(triangles)i(with)h(lar)-66 b(gest)464 b(side)h(n.)597 b(Se)-66 b(e)563 b FL([)p 0 1 0 0 TeXcolorcmyk(22)p (#cite.jm00amm) [[513 81 524 93] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)p 0 1 0 0 TeXcolorcmyk 433 w(23)p (#cite.m74amm) [[531 81 543 93] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p Black 26475 74617 a(4)p Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black Black Black 1675 1424 a FJ(23.)p Black 650 w(The)430 b(value)h(of)g FG(f)142 b FL(\()p FG(n)p FL(\))431 b FJ(wher)-66 b(e)430 b FG(f)573 b FJ(is)430 b(the)h(solution)g(of)f(the)h(functional)e(e)-66 b(quation)430 b FG(f)142 b FL(\()p FG(m)219 b FL(+)g FG(k)45 b FL(\))219 b FH(\241)g FG(f)142 b FL(\()p FG(m)219 b FH(\241)4052 3350 y FG(k)45 b FL(\))369 b(=)f FG(k)45 b FL(\()p FG(m)295 b FL(+)g(1\))465 b FJ(for)f(p)-66 b(ositive)464 b(inte)-66 b(gers)463 b FG(k)414 b(<)368 b(m)p FJ(,)465 b(and)f FG(f)142 b FL(\(1\))369 b(=)g(1)p FG(;)221 b(f)142 b FL(\(2\))370 b(=)e(2)p FJ(.)p Black 1675 6383 a(24.)p Black 650 w(The)464 b FG(n)7484 5901 y FF(th)8887 6383 y FJ(term)g(of)h(the)f(r)-66 b(ow)465 b FL(3)370 b(\()p FJ(and)464 b(c)-66 b(olumn)465 b FL(3\))g FJ(of)f(L)-66 b(osanitsch's)464 b(arr)-66 b(ay.)p 1972 7424 50353 45 v 1972 27190 45 19766 v 5005 8752 a(L)g(osanitsch's)464 b(arr)-66 b(ay,)464 b(values)i(of)e FG(L)p FL(\()p FG(r)-36 b(;)221 b(c)p FL(\))465 b FJ(fr)-66 b(om)563 b FL([)p 0 1 0 0 TeXcolorcmyk(32)p (#cite.class) [[335 638 347 650] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])2348 10543 y FG(r)36 b FH(n)221 b FG(c)p 4947 11121 45 1927 v 1108 w FJ(1)1107 b(2)1439 b(3)1771 b(4)h(5)2103 b(6)2436 b(7)f(8)h(9)f(10)h(11)4672 b(se)-66 b(q.)596 b(no.)h(in)464 b(OEIS)626 b FL([)p 0 1 0 0 TeXcolorcmyk(31)p (#cite.OEIS) [[521 622 533 634] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 2348 11165 49248 45 v 3050 12514 a FJ(1)p 4947 13091 45 1927 v 1809 w(1)1107 b(1)1439 b(1)1771 b(1)h(1)2103 b(1)2436 b(1)f(1)h(1)2767 b(1)3100 b(1)2103 b(.)199 b(.)g(.)1306 b(A000012)3050 14440 y(2)p 4947 15018 V 1809 w(1)1107 b(1)1439 b(2)1771 b(2)h(3)2103 b(3)2436 b(4)f(4)h(5)2767 b(5)3100 b(6)2103 b(.)199 b(.)g(.)p 0 1 0 0 TeXcolorcmyk 1306 w(A004526)p [[425 587 469 599] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=004526) pdfm Black 3050 16366 a(3)p 4947 16944 V 1809 w(1)1107 b(2)1439 b(4)1771 b(6)h(9)f(12)h(16)f(20)h(25)2103 b(30)2436 b(36)1771 b(.)199 b(.)g(.)p 0 1 0 0 TeXcolorcmyk 1306 w Fw(A002620)p [[425 570 478 582] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002620) pdfm Black 3050 18292 a FJ(4)p 4947 18870 V 1809 w(1)1107 b(2)1439 b(6)g(10)1108 b(19)1439 b(28)1772 b(44)f(60)h(85)f(110)h(146) 1439 b(.)199 b(.)g(.)p 0 1 0 0 TeXcolorcmyk 1306 w(A005993)p [[425 552 469 564] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005993) pdfm Black 3050 20218 a(5)p 4947 20796 V 1809 w(1)1107 b(3)1439 b(9)g(19)1108 b(38)1439 b(66)g(110)1108 b(170)f(255)1440 b(365)1772 b(511)1439 b(.)199 b(.)g(.)p 0 1 0 0 TeXcolorcmyk 1306 w(A005994)p [[425 535 469 547] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005994) pdfm Black 3050 22144 a(6)p 4947 22722 V 1809 w(1)1107 b(3)g(12)g(28)h(66)f (126)g(236)h(396)f(651)h(1001)g(1512)f(.)199 b(.)g(.)p 0 1 0 0 TeXcolorcmyk 1306 w(A005995)p [[425 518 469 530] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005995) pdfm Black 2348 24673 a FG(L)p FL(\()p FG(r)-36 b(;)221 b(c)p FL(\))369 b(=)g FG(L)p FL(\()p FG(r)-36 b(;)221 b(c)254 b FH(\241)g FL(1\))g(+)f FG(L)p FL(\()p FG(r)290 b FH(\241)254 b FL(1)p FG(;)221 b(c)p FL(\))p FH(\241)21913 23328 y Fz(n)22925 24166 y Fs(\()23396 23663 y FC(\()p FA(r)22 b FC(+)p FA(c)p FC(\))p FA(=)p FC(2)24230 24447 y FA(c=)p FC(2)26265 24166 y Fs(\))26736 24084 y FF(;)345 b Fy(if)339 b(b)-49 b(oth)340 b FF(r)-26 b(;)157 b(c)339 b Fy(even)24595 25403 y FD(0)p FF(;)3253 b Fy(otherwise)34026 24673 y FJ(and)447 b FG(L)p FL(\(1)p FG(;)221 b(c)p FL(\))369 b(=)f FG(L)p FL(\()p FG(r)-36 b(;)221 b FL(1\))370 b(=)f(1)446 b FJ(for)g(al)66 b(l)2348 26599 y FG(r)-36 b(;)221 b(c)466 b FJ(p)-66 b(ositive)463 b(inte)-66 b(gers.)p 52280 27190 45 19766 v 1972 27234 50353 45 v Black 1675 29581 a(25.)p Black 650 w FL(1)295 b(+)g FH(j)p FG(A)7648 29780 y FF(n)8274 29581 y FH(j)1794 b FJ(wher)-66 b(e)464 b FG(A)15081 29780 y FF(n)16076 29581 y FL(=)17457 28505 y FE(\251)18453 29581 y FH(f)p FG(i;)221 b(j)75 b FH(g)370 b(\265)f(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)p FH(g)371 b(j)590 b FG(i)369 b FH(6)p FL(=)f FG(j)540 b FJ(and)465 b FG(n)369 b FH(\267)g FG(i)295 b FL(+)g FG(j)41411 28505 y FE(\252)4052 32060 y FJ(this)517 b(is)h(one)g(mor)-66 b(e)517 b(than)h(the)f(sum)i (for)f FG(n)467 b FH(\267)h FG(m)f FH(\267)g FL(2)p FG(n)335 b FH(\241)g FL(1)518 b FJ(of)g(the)g(numb)-66 b(er)516 b(of)i(p)-66 b(artitions)517 b(of)g(m)4052 33986 y(with)464 b(two)i(distinct)d(p)-66 b(arts)465 b(fr)-66 b(om)464 b FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)p FH(g)p FJ(.)p Black 1675 37020 a(26.)p Black 650 w(The)464 b(sum)i(of)e(the)h FG(n)13927 36537 y FF(th)15330 37020 y FJ(r)-66 b(ow)465 b(of)f(the)h(fol)66 b(lowing)465 b(arr)-66 b(ay.)4716 39290 y FG(n)p FH(n)221 b FG(k)p 7741 39868 45 1927 v 1435 w FJ(1)1449 b(2)h(3)f(4)h(5)1389 b(6)1329 b(7)f(8)h(9)p 4052 39912 23578 45 v 5575 41261 a(1)p 7741 41838 45 1927 v 2188 w Fw(1)5575 43187 y FJ(2)p 7741 43765 V 2249 w(1)1449 b(1)5575 45113 y(3)p 7741 45691 V 2249 w(1)1389 b Fw(2)g FJ(1)5575 47039 y(4)p 7741 47617 V 2249 w(1)1449 b(2)h(2)f(1)5575 48965 y(5)p 7741 49543 V 2249 w(1)g(2)1389 b Fw(3)g FJ(2)1450 b(1)5575 50891 y(6)p 7741 51469 V 2249 w(1)f(2)h(3)f(3)h(2)1389 b(1)5575 52817 y(7)p 7741 53395 V 2249 w(1)1449 b(2)h(3)1389 b Fw(4)g FJ(3)g(2)1329 b(1)5575 54743 y(8)p 7741 55321 V 2249 w(1)1449 b(2)h(3)f(4)h(4)1389 b(3)1329 b(2)f(1)5575 56669 y(9)p 7741 57247 V 2249 w(1)1449 b(2)h(3)f(4)1389 b Fw(5)1329 b FJ(4)g(3)f(2)h(1)p Black 1675 59387 a(27.)p Black 650 w(One)442 b(mor)-66 b(e)443 b(than)f(the)h(sum)g(for)g FG(n)370 b FH(\267)f FG(m)f FH(\267)i FL(2)p FG(n)247 b FH(\241)g FL(1)443 b FJ(of)g(the)f(numb)-66 b(er)442 b(of)h(p)-66 b(artitions)441 b(of)i(m)g(with)g(two)4052 62120 y(p)-66 b(arts)424 b(minus)h FG(n)206 b FH(\241)g FL(1)425 b FJ(cho)-66 b(ose)424 b(2)1754 b FL(=)1697 b(1)206 b(+)25669 60460 y FD(2)p FF(n)p FB(\241)p FD(1)25831 60858 y FE(X)25722 63648 y FF(m)p FD(=)p FF(n)28133 60247 y FE(\271)29041 61222 y FG(m)295 b FH(\241)g FL(1)p 29041 61815 3412 54 v 30421 63032 a(2)32585 60247 y FE(\272)33566 62120 y FH(\241)34805 60247 y FE(\265)35783 61222 y FG(n)g FH(\241)h FL(1)36983 63032 y(2)38833 60247 y FE(\266)41508 62120 y FL(=)1697 b(1)206 b(+)46291 60460 y FD(2)p FF(n)p FB(\241)p FD(1)46453 60858 y FE(X)46345 63648 y FF(m)p FD(=)p FF(n)48756 60646 y FE(j)49590 61222 y FG(m)p 49590 61815 1138 54 v 49833 63032 a FL(2)50860 60646 y FE(k)51767 62120 y FH(\241)4052 64343 y FE(j)4886 64919 y FG(n)p 4886 65512 777 54 v 4949 66729 a FL(2)5795 64343 y FE(k)6791 65818 y FH(\241)8119 63945 y FE(\265)9097 64919 y FG(n)296 b FH(\241)f FL(1)10297 66729 y(2)12148 63945 y FE(\266)13125 65818 y FG(;)4052 69554 y FL(=)1697 b(1)295 b(+)9086 67894 y FF(n)p FB(\241)p FD(1)9013 68292 y FE(X)9211 71092 y FF(i)p FD(=0)11153 67681 y FE(\271)12061 68656 y FG(n)g FH(\241)h FL(1)f(+)g FG(i)p 12061 69249 5096 54 v 14284 70465 a FL(2)17290 67681 y FE(\272)18360 69554 y FH(\241)19688 67681 y FE(\265)20666 68656 y FG(n)g FH(\241)h FL(1)21866 70465 y(2)23716 67681 y FE(\266)26391 69554 y FL(=)1697 b(1)296 b(+)31425 67894 y FF(n)p FB(\241)p FD(1)31353 68292 y FE(X)31551 71092 y FF(i)p FD(=0)33493 67681 y FE(\273)34400 68656 y FG(n)g FH(\241)f FL(2)h(+)f FG(i)p 34400 69249 V 36623 70465 a FL(2)39629 67681 y FE(\274)40699 69554 y FH(\241)42028 67681 y FE(\265)43006 68656 y FG(n)g FH(\241)g FL(1)44205 70465 y(2)46056 67681 y FE(\266)47034 69554 y FG(;)p Black 26475 74617 a FL(5)p Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black Black 4052 2819 a FL(=)6761 96 y FE(8)6761 1291 y(<)6761 3682 y(:)7941 1793 y FG(f)8582 1992 y FF(f)9188 1793 y FL(\()p FG(n)p FL(\))295 b(+)g FG(n;)1329 b FJ(if)464 b FG(n)h FJ(o)-66 b(dd)7941 4105 y FG(f)8582 4304 y FF(f)9188 4105 y FL(\()p FG(n)p FL(\))p FG(;)3707 b FJ(if)464 b FG(n)h FJ(even)21997 2819 y(wher)-66 b(e)465 b FG(f)26308 3018 y FF(f)26913 2819 y FL(\()p FG(n)p FL(\))369 b(=)f(\()p FG(n)296 b FL(+)e FH(b)p FG(n=)p FL(2)p FH(c)p FL(\))p FH(b)p FG(n=)p FL(2)p FH(c)299 b(\241)41979 1743 y FE(\241)42588 2229 y FF(n)42639 3277 y FD(2)43159 1743 y FE(\242)43768 2819 y FJ(,)4052 7712 y FL(=)6761 4989 y FE(8)6761 6184 y(<)6761 8575 y(:)7941 6686 y FG(f)8582 6885 y FF(c)9045 6686 y FL(\()p FG(n)p FL(\))d FH(\241)f FG(n;)1329 b FJ(if)464 b FG(n)h FJ(o)-66 b(dd)7941 8998 y FG(f)8582 9197 y FF(c)9045 8998 y FL(\()p FG(n)p FL(\))p FG(;)3729 b FJ(if)464 b FG(n)h FJ(even)21876 7712 y(wher)-66 b(e)465 b FG(f)26187 7911 y FF(c)26650 7712 y FL(\()p FG(n)p FL(\))369 b(=)g(\()p FG(n)295 b FL(+)g FH(d)p FG(n=)p FL(2)p FH(e)p FL(\))p FH(d)p FG(n=)p FL(2)p FH(e)j(\241)41717 6636 y FE(\241)42326 7122 y FF(n)42376 8170 y FD(2)42897 6636 y FE(\242)43505 7712 y FJ(.)p Black 1675 11762 a(28.)p Black 650 w(T)-100 b(ur\266)-664 b(an)-133 b('s)488 b(numb)-66 b(er)487 b(for)g(triangles)g(in)h(a)g(gr)-66 b(aph)488 b(on)f FG(n)313 b FL(+)f(1)488 b FJ(vertic)-66 b(es)486 b FL(=)i FJ(the)g(maximum)f(numb)-66 b(er)486 b(of)4052 13689 y(e)-66 b(dges)434 b(of)g(a)h(gr)-66 b(aph)435 b(on)f FG(n)229 b FL(+)f(1)435 b FJ(vertic)-66 b(es)433 b(which)i(has)g(no)g (triangles)e FL(=)37734 12613 y FE(\241)38343 13099 y FF(n)p FD(+1)38994 14147 y(2)40116 12613 y FE(\242)40953 13689 y FH(\241)42214 12613 y FE(\241)42823 13084 y FB(b)43374 12711 y FA(n)p FC(+1)p 43374 12883 1539 40 v 43940 13431 a(2)45045 13084 y FB(c)43908 14162 y FD(2)45463 12613 y FE(\242)46300 13689 y FH(\241)47562 12613 y FE(\241)48170 13084 y FB(b)48721 12711 y FA(n)p FC(+2)p 48721 12883 V 49288 13431 a(2)50392 13084 y FB(c)49255 14162 y FD(2)50811 12613 y FE(\242)51788 13689 y FL(=)4052 14539 y FE(\241)4661 15025 y FF(n)p FD(+1)5312 16073 y(2)6434 14539 y FE(\242)7338 15615 y FH(\241)8666 14539 y FE(\241)9275 15010 y FB(d)9826 14648 y FA(n)p 9826 14809 512 40 v 9879 15358 a FC(2)10470 15010 y FB(e)9846 16088 y FD(2)10888 14539 y FE(\242)11792 15615 y FH(\241)13121 14539 y FE(\241)13729 15010 y FB(d)14280 14637 y FA(n)p FC(+1)p 14280 14809 1539 40 v 14847 15358 a(2)15951 15010 y FB(e)14814 16088 y FD(2)16370 14539 y FE(\242)17347 15615 y FL(=)18728 14539 y FE(\241)19337 15010 y FB(b)19888 14637 y FA(n)p FC(+2)p 19888 14809 V 20454 15358 a(2)21559 15010 y FB(c)20422 16088 y FD(2)21977 14539 y FE(\242)22881 15615 y FL(+)24188 14539 y FE(\241)24797 15010 y FB(b)25348 14637 y FA(n)p FC(+3)p 25348 14809 V 25914 15358 a(2)27019 15010 y FB(c)25882 16088 y FD(2)27437 14539 y FE(\242)28415 15615 y FL(=)29795 14539 y FE(\241)30404 15010 y FB(d)30955 14637 y FA(n)p FC(+1)p 30955 14809 V 31521 15358 a(2)32626 15010 y FB(e)31489 16088 y FD(2)33045 14539 y FE(\242)33949 15615 y FL(+)35255 14539 y FE(\241)35864 15010 y FB(d)36415 14637 y FA(n)p FC(+2)p 36415 14809 V 36981 15358 a(2)38086 15010 y FB(e)36949 16088 y FD(2)38504 14539 y FE(\242)39482 15615 y FL(=)40863 14539 y FE(\241)41472 15010 y FB(b)42023 14637 y FA(n)p FC(+2)p 42023 14809 V 42589 15358 a(2)43694 15010 y FB(c)42557 16088 y FD(2)44112 14539 y FE(\242)45016 15615 y FL(+)46323 14539 y FE(\241)46932 15010 y FB(d)47483 14637 y FA(n)p FC(+2)p 47483 14809 V 48049 15358 a(2)49154 15010 y FB(e)48017 16088 y FD(2)49572 14539 y FE(\242)50181 15615 y FJ(.)p Black 1675 18499 a(29.)p Black 1631 w FL(Max)4052 19401 y FF(u)p FB(2)p FD([0)p FF(;)p FD(1])6947 19149 y FA(n)p FC(+1)11027 17503 y FE(P)9133 19645 y FD(1)p FB(\267)p FF(i)h(n)295 b FL(+)g(1)434 b(and)f(the)g(de\257nition)g(fails.)800 12577 y FK(B.)490 b FL(Let)f(\255)6092 12776 y FF(n)7183 12577 y FL(=)464 b FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)p FH(g)p FG(;)g(f)142 b FL(\()p FG(i)p FL(\))465 b(=)f FG(i)490 b FL(and)f FG(g)48 b FL(\()p FG(i)p FL(\))463 b(=)g FH(d)27395 12054 y FF(n)p 27395 12271 571 54 v 27520 13035 a(i)28099 12577 y FH(e)490 b FL(for)g(1)465 b FH(\267)f FG(i)g FH(\267)h FG(n)p FL(.)746 b(If)490 b(5)465 b FH(\267)f FG(n)p FL(,)504 b FG(f)632 b FL(and)489 b FG(g)537 b FL(are)490 b FJ(not)800 14673 y FL(an)-36 b(tagonistic,)435 b(since)895 b(Max)11617 15559 y FF(i)p FB(2f)p FD(1)p FF(::n)p FB(g)15291 14673 y FG(f)142 b FL(\()p FG(i)p FL(\))p FH(\242)p FG(g)48 b FL(\()p FG(i)p FL(\))367 b FG(<)21774 13199 y FE(j)22475 13597 y(\241)23217 14150 y FF(n)p FD(+1)p 23217 14368 1774 54 v 23868 15131 a(2)25123 13597 y FE(\242)25732 13895 y FD(2)26257 13199 y FE(k)27392 14673 y FL(and)433 b(the)g(de\257nition)g(fails.)800 18923 y Fr(2.2)1793 b(Prop)50 b(erties)599 b(of)f(an)-50 b(tagonistic)598 b(functions)p Black 800 21707 a FK(Prop)42 b(osition)500 b(2.2)p Black 651 w FJ(L)-66 b(et)545 b FG(n)h FJ(b)-66 b(e)546 b(a)g(p)-66 b(ositive)545 b(inte)-66 b(ger,)563 b FL(\255)547 b FJ(a)f(\257nite)f(set,)566 b FG(f)688 b FJ(and)546 b FG(g)594 b FJ(functions)544 b(fr)-66 b(om)546 b FL(\255)g FJ(to)800 23633 y FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)p FH(g)p FJ(,)466 b(such)g(that)e(for)h(every)f FG(u)369 b FH(2)g FL(\255)p FG(;)221 b(f)142 b FL(\()p FG(u)p FL(\))296 b(+)f FG(g)48 b FL(\()p FG(u)p FL(\))368 b FH(\267)h FG(n)296 b FL(+)f(1)p FJ(,)464 b(then)800 25559 y FG(f)673 b FJ(and)532 b FG(g)578 b FJ(ar)-66 b(e)531 b(antagonistic)f(of)h(or)-66 b(der)531 b FG(n)g FJ(if)g(and)g(only)g(if)g(ther)-66 b(e)530 b(is)h(a)h FG(w)527 b FH(2)492 b FL(\255)531 b FJ(such)h(that)46090 24085 y FE(j)46792 24483 y(\241)47533 25036 y FF(n)p FD(+1)p 47533 25254 V 48185 26017 a(2)49439 24483 y FE(\242)50048 24781 y FD(2)50574 24085 y FE(k)51767 25559 y FH(\267)800 27485 y FG(f)142 b FL(\()p FG(w)36 b FL(\))294 b FH(\242)h FG(g)48 b FL(\()p FG(w)36 b FL(\))p FJ(.)800 30850 y FK(Pro)42 b(of)510 b FL(There)374 b(exists)h FG(w)405 b FH(2)368 b FL(\255)375 b(suc)-36 b(h)373 b(that)h FG(f)142 b FL(\()p FG(w)36 b FL(\))174 b FH(\242)g FG(g)48 b FL(\()p FG(w)36 b FL(\))367 b FH(\270)29596 29376 y FE(j)30297 29774 y(\241)31039 30327 y FF(n)p FD(+1)p 31039 30545 V 31690 31308 a(2)32945 29774 y FE(\242)33554 30072 y FD(2)34080 29376 y FE(k)35155 30850 y FL(is)375 b(the)e(same)i(as)g (Max)43660 31677 y FF(u)p FB(2)p FD(\255)46072 30850 y FG(f)142 b FL(\()p FG(u)p FL(\))p FH(\242)p FG(g)48 b FL(\()p FG(u)p FL(\))368 b FH(\270)800 32029 y FE(j)1501 32428 y(\241)2243 32981 y FF(n)p FD(+1)p 2243 33198 V 2894 33962 a(2)4149 32428 y FE(\242)4758 32725 y FD(2)5283 32029 y FE(k)6529 33504 y FL(and)545 b(the)f(opp)36 b(osite)545 b(inequalit)-36 b(y)546 b(follo)-36 b(ws)547 b(from)e(the)f(AM-GM)g (inequalit)-36 b(y)546 b FG(ab)558 b FH(\267)47755 32029 y FE(j)48456 32428 y(\241)49197 32981 y FF(a)p FD(+)p FF(b)p 49197 33198 1635 54 v 49779 33962 a FD(2)50964 32428 y FE(\242)51573 32716 y FD(2)52099 32029 y FE(k)800 35430 y FL(and)433 b(the)g(assumption)g FG(f)142 b FL(\()p FG(u)p FL(\))295 b(+)g FG(g)48 b FL(\()p FG(u)p FL(\))368 b FH(\267)i FG(n)295 b FL(+)g(1.)579 b Fq(\244)p Black 800 38795 a FK(Lemma)499 b(2.3)p Black 651 w FJ(L)-66 b(et)464 b FG(i)g FJ(and)h FG(j)540 b FJ(b)-66 b(e)464 b(p)-66 b(ositive)464 b(inte)-66 b(gers,)462 b(then)i FH(j)p FG(i)295 b FH(\241)g FG(j)75 b FH(j)369 b(\267)g FL(1)739 b FH(\()-221 b(\))39555 37321 y FE(j)40389 38162 y FD(\()p FF(i)p FD(+)p FF(j)51 b FD(\))42605 37849 y FC(2)p 40389 38490 2677 54 v 41492 39253 a FD(4)43198 37321 y FE(k)44269 38795 y FH(\267)369 b FG(i)p FH(\242)o FG(j)800 42160 y FK(Pro)42 b(of.)820 b FL(Let)514 b FG(i)g FL(and)g FG(j)589 b FL(b)36 b(e)514 b(p)36 b(ositiv)-36 b(e)515 b(in)-36 b(tegers,)535 b FH(j)p FG(i)350 b FH(\241)g FG(j)75 b FH(j)507 b(\267)f FL(1)1013 b FH(\()-221 b(\))1014 b FL(\()p FG(i)349 b FH(\241)i FG(j)75 b FL(\))39272 41678 y FD(2)40303 42160 y FH(\267)507 b FL(1)1013 b FH(\()-221 b(\))1013 b FL(\()p FG(i)350 b FH(\241)h FG(j)75 b FL(\))50757 41678 y FD(2)51788 42160 y FG(<)800 44086 y FL(4)738 b FH(\()-221 b(\))739 b FL(\()p FG(i)279 b FL(+)h FG(j)75 b FL(\))9002 43604 y FD(2)9896 44086 y FG(<)369 b FL(4\()p FG(ij)355 b FL(+)279 b(1\))739 b FH(\()-221 b(\))20262 43453 y FD(\()p FF(i)p FD(+)p FF(j)51 b FD(\))22478 43141 y FC(2)p 20262 43781 V 21365 44545 a FD(4)23351 44086 y FH(\241)281 b FL(1)369 b FG(<)g(ij)812 b FH(\()-221 b(\))738 b(b)32755 43453 y FD(\()p FF(i)p FD(+)p FF(j)51 b FD(\))34971 43141 y FC(2)p 32756 43781 V 33859 44545 a FD(4)35565 44086 y FH(c)370 b(\267)f FG(ij)75 b FL(,)427 b(for)g(the)f(last)g(implication)800 46013 y(see)434 b([)p 0 1 0 0 TeXcolorcmyk(20)p (#cite.gkp89) [[101 303 113 315] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)g(p.69].)579 b Fq(\244)p Black 800 49378 a FK(F)-125 b(act)500 b(2.4)p Black 651 w FJ(The)464 b(function)g FG(m)368 b FH(7!)i(b)18343 48855 y FF(m)19175 48542 y FC(2)p 18343 49072 1294 54 v 18754 49836 a FD(4)19769 49378 y FH(c)465 b FJ(on)g(the)g(p)-66 b(ositive)463 b(inte)-66 b(gers)463 b(is)p Black 2338 51943 a(1.)p Black 651 w(strictly)h(incr)-66 b(e)g(asing)462 b(and)j(thus)g(is)g (one-to-one,)e(and)p Black 2338 54976 a(2.)p Black 651 w FH(b)4775 54453 y FF(m)5607 54140 y FC(2)p 4775 54670 V 5186 55434 a FD(4)6201 54976 y FH(c)370 b(\267)f(b)9286 54453 y FF(n)9857 54140 y FC(2)p 9286 54670 1033 54 v 9567 55434 a FD(4)10451 54976 y FH(c)g FL(=)-221 b FH(\))369 b FG(m)g FH(\267)g FG(n)465 b FJ(for)g(al)66 b(l)465 b FG(m)g FJ(and)g FG(n)g FJ(p)-66 b(ositive)464 b(inte)-66 b(gers.)p Black 800 58341 a FK(Lemma)499 b(2.5)p Black 651 w FJ(L)-66 b(et)713 b FG(n)i FJ(b)-66 b(e)714 b(a)h(p)-66 b(ositive)713 b(inte)-66 b(ger,)774 b FL(\255)715 b FJ(a)g(\257nite)e (set,)776 b FG(f)857 b FJ(and)714 b FG(g)762 b FJ(functions)714 b(fr)-66 b(om)713 b FL(\255)i FJ(to)800 60267 y FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)p FH(g)p FJ(,)466 b(such)g(that)e(for)h(every)f FG(u)369 b FH(2)g FL(\255)p FG(;)221 b(f)142 b FL(\()p FG(u)p FL(\))296 b(+)f FG(g)48 b FL(\()p FG(u)p FL(\))368 b FH(\267)h FG(n)296 b FL(+)f(1)p FJ(,)464 b(then)h(for)f(every)g FG(w)405 b FH(2)368 b FL(\255)p FJ(,)2751 61051 y FE(j)3585 61892 y FD(\()p FF(n)p FD(+1\))6090 61580 y FC(2)p 3585 62220 2967 54 v 4833 62983 a FD(4)6684 61051 y FE(k)7754 62525 y FH(\267)h FG(f)142 b FL(\()p FG(w)36 b FL(\))295 b FH(\242)g FG(g)48 b FL(\()p FG(w)36 b FL(\))1327 b FJ(if)464 b(and)h(only)f(if)1328 b FG(f)142 b FL(\()p FG(w)36 b FL(\))294 b(+)h FG(g)48 b FL(\()p FG(w)36 b FL(\))367 b(=)i FG(n)296 b FL(+)e(1)835 b FJ(and)e FH(j)p FG(f)142 b FL(\()p FG(w)36 b FL(\))295 b FH(\241)g FG(g)48 b FL(\()p FG(w)36 b FL(\))p FH(j)368 b(\267)h FL(1)p FG(:)800 66466 y FK(Pro)42 b(of.)792 b FL(\()p FH(\))505 b FJ(left)529 b(p)-66 b(art)126 b FL(\))505 b(By)g(AM-GM,)21430 64992 y FE(j)22264 65833 y FD(\()p FF(n)p FD(+1\))24769 65520 y FC(2)p 22264 66161 V 23512 66924 a FD(4)25363 64992 y FE(k)26554 66466 y FH(\267)490 b FG(f)142 b FL(\()p FG(w)36 b FL(\))343 b FH(\242)h FG(g)48 b FL(\()p FG(w)36 b FL(\))489 b FH(\))36850 64992 y FE(j)37684 65833 y FD(\()p FF(n)p FD(+1\))40189 65520 y FC(2)p 37684 66161 V 38932 66924 a FD(4)40783 64992 y FE(k)41974 66466 y FH(\267)43498 64992 y FE(j)44332 65833 y FD(\()p FF(f)98 b FD(\()p FF(w)24 b FD(\)+)p FF(g)32 b FD(\()p FF(w)24 b FD(\)\))49688 65520 y FC(2)p 44332 66161 5816 54 v 47004 66924 a FD(4)50280 64992 y FE(k)51472 66466 y FH(\))800 68392 y FG(n)276 b FL(+)f(1)369 b FH(\267)g FG(f)142 b FL(\()p FG(w)36 b FL(\))274 b(+)h FG(g)48 b FL(\()p FG(w)36 b FL(\))423 b(the)g(last)h(b)-36 b(y)424 b(fact)p 0 1 0 0 TeXcolorcmyk 425 w(2.4)p (#fact.2.4) [[270 101 285 113] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)j(and)c(since)h FG(f)142 b FL(\()p FG(w)36 b FL(\))274 b(+)h FG(g)48 b FL(\()p FG(w)36 b FL(\))368 b FH(\267)h FG(n)275 b FL(+)g(1)424 b(b)-36 b(y)424 b(assumption,)i(w) -36 b(e)800 70318 y(get)434 b FG(f)142 b FL(\()p FG(w)36 b FL(\))294 b(+)h FG(g)48 b FL(\()p FG(w)36 b FL(\))367 b(=)i FG(n)296 b FL(+)e(1.)p Black 26475 74617 a(7)p Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black Black 800 1623 a FL(\()p FJ(right)380 b(p)-66 b(art)126 b FL(\))342 b FG(f)142 b FL(\()p FG(w)36 b FL(\))109 b(+)g FG(g)48 b FL(\()p FG(w)36 b FL(\))366 b FH(\267)j FG(n)109 b FL(+)g(1)343 b(and)21388 149 y FE(j)22222 990 y FD(\()p FF(n)p FD(+1\))24727 678 y FC(2)p 22222 1318 2967 54 v 23469 2081 a FD(4)25321 149 y FE(k)26391 1623 y FH(\267)369 b FG(f)142 b FL(\()p FG(w)36 b FL(\))109 b FH(\242)g FG(g)48 b FL(\()p FG(w)36 b FL(\))366 b FH(\))35854 149 y FE(j)36687 990 y FD(\()p FF(f)98 b FD(\()p FF(w)24 b FD(\)+)p FF(g)32 b FD(\()p FF(w)24 b FD(\)\))42043 678 y FC(2)p 36687 1318 5816 54 v 39360 2081 a FD(4)42636 149 y FE(k)43706 1623 y FH(\267)369 b FG(f)142 b FL(\()p FG(w)36 b FL(\))109 b FH(\242)g FG(g)48 b FL(\()p FG(w)36 b FL(\))367 b FH(\))800 3549 y(j)p FG(f)142 b FL(\()p FG(w)36 b FL(\))294 b FH(\241)i FG(g)48 b FL(\()p FG(w)36 b FL(\))p FH(j)367 b(\267)i FL(1)434 b(b)-36 b(y)434 b(lemma)p 0 1 0 0 TeXcolorcmyk 434 w(2.3)p (#fact.2.3) [[229 685 244 697] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)579 b Fq(\244)800 5476 y FK(Pro)42 b(of.)1130 b FL(\()p FH(\()p FL(\))618 b(\()p FJ(this)634 b(is)g(use)-66 b(d)634 b(sever)-66 b(al)634 b(times)f(in)g(the)h(fol)66 b(lowing)634 b(pr)-66 b(o)g(of)633 b(of)h(the)g(main)f(the)-66 b(or)g(em)99 b FL(\))617 b(By)800 7402 y(lemma)p 0 1 0 0 TeXcolorcmyk 419 w(2.3)p (#fact.2.3) [[117 650 132 662] [1 1 1 [3 3]] [0 0 1]] pdfm Black 419 w FH(j)p FG(f)142 b FL(\()p FG(w)36 b FL(\))264 b FH(\241)g FG(g)48 b FL(\()p FG(w)36 b FL(\))p FH(j)368 b(\267)h FL(1)g FH(\))19252 5927 y FE(j)20086 6769 y FD(\()p FF(f)98 b FD(\()p FF(w)24 b FD(\)+)p FF(g)32 b FD(\()p FF(w)24 b FD(\)\))25442 6456 y FC(2)p 20086 7096 V 22758 7860 a FD(4)26034 5927 y FE(k)27104 7402 y FH(\267)369 b FG(f)142 b FL(\()p FG(w)36 b FL(\))264 b FH(\242)g FG(g)48 b FL(\()p FG(w)36 b FL(\),)420 b(but)e(since)g FG(f)142 b FL(\()p FG(w)36 b FL(\))264 b(+)g FG(g)48 b FL(\()p FG(w)36 b FL(\))367 b(=)i FG(n)265 b FL(+)f(1)800 9903 y(w)-36 b(e)434 b(get)4883 8429 y FE(j)5717 9270 y FD(\()p FF(n)p FD(+1\))8222 8958 y FC(2)p 5717 9598 2967 54 v 6965 10362 a FD(4)8816 8429 y FE(k)9886 9903 y FH(\267)369 b FG(f)142 b FL(\()p FG(w)36 b FL(\))294 b FH(\242)h FG(g)48 b FL(\()p FG(w)36 b FL(\).)577 b Fq(\244)800 12494 y FL(In)433 b(summary)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing.)p Black 800 15527 a FK(Prop)42 b(osition)500 b(2.6)e(\(Characterization)k(of)c (an)-42 b(tagonistic)502 b(functions\))p Black 652 w FJ(L)-66 b(et)439 b FG(n)i FJ(b)-66 b(e)439 b(a)i(p)-66 b(ositive)439 b(in-)800 17453 y(te)-66 b(ger,)394 b FL(\255)380 b FJ(a)f(\257nite)f(set,)395 b(and)379 b FG(f)522 b FJ(and)379 b FG(g)426 b FJ(functions)378 b(fr)-66 b(om)378 b FL(\255)i FJ(to)f FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)p FH(g)380 b FJ(such)g(that)f FG(f)142 b FL(\()p FG(u)p FL(\))105 b(+)g FG(g)48 b FL(\()p FG(u)p FL(\))367 b FH(\267)i FG(n)105 b FL(+)g(1)800 19379 y FJ(for)411 b(al)66 b(l)413 b FG(u)369 b FH(2)f FL(\255)p FJ(,)423 b(then)410 b FG(f)554 b FJ(and)411 b FG(g)459 b FJ(ar)-66 b(e)412 b(antagonistic)d(of)i(or)-66 b(der)412 b FG(n)g FJ(on)f FL(\255)h FJ(if)f(and)g(only)g(if)g(ther)-66 b(e)411 b(exists)g FG(w)404 b FH(2)369 b FL(\255)800 21305 y FJ(such)466 b(that)e FG(f)142 b FL(\()p FG(w)36 b FL(\))295 b(+)f FG(g)48 b FL(\()p FG(w)36 b FL(\))368 b(=)h FG(n)295 b FL(+)g(1)465 b FJ(and)g FH(j)p FG(f)142 b FL(\()p FG(w)36 b FL(\))294 b FH(\241)i FG(g)48 b FL(\()p FG(w)36 b FL(\))p FH(j)367 b(\267)j FL(1)p FJ(.)2751 24338 y FL(Note)460 b(that,)467 b FH(j)p FG(f)142 b FL(\()p FG(w)36 b FL(\))312 b FH(\241)i FG(g)48 b FL(\()p FG(w)36 b FL(\))p FH(j)412 b(\267)i FL(1)461 b(can)f(b)36 b(e)460 b(replaced)f(b)-36 b(y)460 b FH(j)p FG(f)142 b FL(\()p FG(w)36 b FL(\))313 b FH(\241)g FG(g)48 b FL(\()p FG(w)36 b FL(\))p FH(j)413 b FL(=)40798 22993 y Fz(n)41809 23775 y FD(0)p FF(;)337 b FD(if)313 b FF(n)h FD(o)26 b(dd)41809 24738 y(1)p FF(;)350 b FD(if)314 b FF(n)f FD(ev)-26 b(en)47275 24338 y FL(and)459 b(those)800 26264 y FG(w)545 b FH(2)509 b FL(\255)517 b(for)g(whic)-36 b(h)516 b(the)f(maxim)-36 b(um)517 b(is)f(ac)-36 b(hiev)g(ed)517 b(are)f(exactly)i(those)e(whic) -36 b(h)516 b(satisfy)h(the)f(righ)-36 b(t)516 b(hand)800 28190 y(conditions.)p Black 800 31556 a FK(F)-125 b(act)500 b(2.7)p Black 651 w FJ(L)-66 b(et)444 b FG(A)i FJ(and)f FG(B)513 b FJ(b)-66 b(e)444 b(\257nite)g(sets,)449 b FG(f)587 b FJ(a)446 b(function)e(fr)-66 b(om)444 b FG(A)i Fw(onto)g FG(B)67 b FJ(,)449 b FG(G)d FJ(a)g(mapping)e(fr)-66 b(om)445 b FG(B)512 b FJ(to)800 33482 y Fp(R)527 b FJ(and)f(for)h(al)66 b(l)527 b FG(a)483 b FH(2)f FG(A)p FJ(,)542 b(let)526 b FG(F)181 b FL(\()p FG(a)p FL(\))482 b(=)h FG(G)p FL(\()p FG(f)142 b FL(\()p FG(a)p FL(\)\))p FJ(,)540 b(then)526 b FL(Max)28717 34308 y FF(a)p FB(2)p FF(A)31119 33482 y FG(F)181 b FL(\()p FG(a)p FL(\))482 b(=)h(Max)36186 34318 y FF(b)p FB(2)p FF(B)38564 33482 y FG(G)p FL(\()p FG(b)p FL(\))1328 b FJ(and)1549 b FL(Min)46313 34308 y FF(a)p FB(2)p FF(A)48588 33482 y FG(F)181 b FL(\()p FG(a)p FL(\))482 b(=)800 35614 y(Min)1046 36450 y FF(b)p FB(2)p FF(B)3297 35614 y FG(G)p FL(\()p FG(b)p FL(\))p FJ(.)2751 39111 y FL(In)488 b(items)p 0 1 0 0 TeXcolorcmyk 489 w(13)p (#Item.30) [[144 365 155 377] [1 1 1 [3 3]] [0 0 1]] pdfm Black 489 w(to)p 0 1 0 0 TeXcolorcmyk 488 w(17)p (#Item.34) [[175 365 186 377] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)503 b(of)489 b(the)f(theorem)g(\255)h(is)g(a)f(complemen)-36 b(ted)488 b(lattice.)743 b(It)489 b(w)-36 b(ould)488 b(b)36 b(e)488 b(in)-36 b(teresting)800 41037 y(to)472 b(study)g(those)g(functions)g FG(f)614 b FL(from)473 b(\255)f(to)g FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)k(;)c(n)p FH(g)473 b FL(suc)-36 b(h)471 b(that)h FG(f)614 b FL(and)p 38336 39955 783 54 v 472 w FG(f)g FL(are)472 b(an)-36 b(tagonistic,)483 b(where)p 800 41881 V 800 42963 a FG(f)142 b FL(\()p FG(u)p FL(\))369 b(=)f FG(f)142 b FL(\()p 6373 42232 741 54 v FG(u)p FL(\).)2751 44889 y(Please)477 b(send)e(to)h(the)f(author)h(other)f(examples)i(of)g(these)e (functions.)706 b(\(There)475 b(are)i(more)f(in)f(graph)800 46815 y(theory)-108 b(,)338 b(consider)315 b(upp)36 b(er)313 b(domination)h(\241,)339 b(irredundance)312 b FG(I)104 b(R)326 b FL([)p 0 1 0 0 TeXcolorcmyk(12)p (#cite.cm89dm) [[359 296 371 308] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)339 b(and)314 b(CO-irredundance)e FG(C)95 b(O)36 b(I)104 b(R)326 b FL([)p 0 1 0 0 TeXcolorcmyk(11)p (#cite.cmm00dm) [[532 296 544 308] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])800 48742 y(n)-36 b(um)g(b)36 b(ers\))2751 51332 y(W)-108 b(e)365 b(could)g(coun)-36 b(t)364 b(those)h(elemen)-36 b(ts)365 b(whic)-36 b(h)365 b(ac)-36 b(hiev)g(e)366 b(the)e(maxim)-36 b(um)366 b(in)f(items)p 0 1 0 0 TeXcolorcmyk 365 w(11)p (#Item.28) [[446 255 458 267] [1 1 1 [3 3]] [0 0 1]] pdfm Black 366 w(to)p 0 1 0 0 TeXcolorcmyk 365 w(18)p (#Item.35) [[475 255 486 267] [1 1 1 [3 3]] [0 0 1]] pdfm Black 366 w(of)g(the)g(main)800 53258 y(theorem.)578 b(Note,)434 b(w)-36 b(e)434 b(m)-36 b(ust)433 b(de\257ne)f(when)h(t)-36 b(w)g(o)434 b(elemen)-36 b(ts)433 b(are)h(di\256eren)-36 b(t.)p Black 2737 56623 a FH(\262)p Black 651 w FL(F)-108 b(or)699 b(items)p 0 1 0 0 TeXcolorcmyk 701 w(14)p (#Item.31) [[166 207 177 219] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)768 b(the)699 b(coun)-36 b(t)700 b(is)g(1)p FG(;)221 b FL(2)p FG(;)g FL(1)p FG(;)g FL(2)p FG(;)g FL(1)p FG(;)g FL(2)p FG(;)g FL(1)p FG(;)g FL(2)p FG(;)g FL(1)g FH(\242)g(\242)g(\242) 832 b FL(=)35564 55278 y Fz(n)36574 56060 y FD(1)p FF(;)338 b FD(if)313 b FF(n)h FD(o)26 b(dd)36574 57023 y(2)p FF(;)351 b FD(if)313 b FF(n)h FD(ev)-26 b(en)42281 56623 y FL(whic)-36 b(h)700 b(is)g(sequence)4052 58549 y(A000034.)p Black 2737 61029 a FH(\262)p Black 651 w FL(F)-108 b(or)433 b(items)p 0 1 0 0 TeXcolorcmyk 434 w(11)p (#Item.28) [[161 168 172 180] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)h(the)f(coun)-36 b(t)433 b(is)h(1)p FG(;)221 b FL(2)p FG(;)g FL(2)p FG(;)g FL(6)p FG(;)g FL(8)p FG(;)g(:)g(:)g(:)p Black 2737 63508 a FH(\262)p Black 651 w FL(F)-108 b(or)433 b(item)p 0 1 0 0 TeXcolorcmyk 433 w(16)p (#Item.33) [[156 145 168 157] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)i(the)e(coun)-36 b(t)433 b(is)h(1)p FG(;)221 b FL(2)p FG(;)g FL(2)p FG(;)g FL(6)p FG(;)g FL(8)p FG(;)g(:)g(:)g(:)p Black 2737 65988 a FH(\262)p Black 651 w FL(F)-108 b(or)433 b(item)p 0 1 0 0 TeXcolorcmyk 433 w(17)p (#Item.34) [[156 123 168 135] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)i(the)e(coun)-36 b(t)433 b(is)h(1)p FG(;)221 b FL(2)p FG(;)g FL(2)p FG(;)g FL(6)p FG(;)g FL(7)p FG(;)g(:)g(:)g(:)p Black 2737 68468 a FH(\262)p Black 651 w FL(F)-108 b(or)433 b(item)p 0 1 0 0 TeXcolorcmyk 433 w(18)p (#Item.35) [[156 101 168 113] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)i(the)e(coun)-36 b(t)433 b(is)h(1)p FG(;)221 b FL(2)p FG(;)g FL(2)p FG(;)g FL(5)p FG(;)g FL(4)p FG(;)g(:)g(:)g(:)p Black 26475 74617 a FL(8)p Black eop %%Page: 9 9 9 8 bop Black 0 TeXcolorgray Black Black 800 1424 a Fr(3)1793 b(Pro)50 b(of)597 b(of)h(the)g(theorem)p 800 2895 51953 45 v 800 10518 45 7624 v 1176 4149 a FL(Most)498 b(of)h(the)e (expressions)i(in)-36 b(v)g(olving)499 b(\260o)36 b(ors)498 b(and)g(ceilings)h(in)f(the)f(theorem)h(ma)-36 b(y)498 b(b)36 b(e)498 b(sho)-36 b(wn)497 b(to)i(b)36 b(e)1176 6075 y(equal)403 b(to)f(item)p 0 1 0 0 TeXcolorcmyk 402 w(2)p (#Item.2) [[154 662 159 674] [1 1 1 [3 3]] [0 0 1]] pdfm Black 402 w(b)-36 b(y)402 b(setting)g FG(n)369 b FL(=)g(2)p FG(k)447 b FL(and)402 b FG(n)369 b FL(=)g(2)p FG(k)276 b FH(\241)231 b FL(1)402 b(and)f(manipulating)h(the)f(resulting)h (algebraic)1176 8001 y(expression.)930 b(Suc)-36 b(h)549 b(examples)i(are)f(items)p 0 1 0 0 TeXcolorcmyk 551 w(3)p (#Item.3) [[280 645 286 657] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 551 w(4)p (#Item.4) [[294 645 300 657] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 551 w(5)p (#Item.5) [[309 645 314 657] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 551 w(6)p (#Item.6) [[323 645 328 657] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 551 w(27)p (#Item.44) [[337 645 348 657] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)581 b(and)p 0 1 0 0 TeXcolorcmyk 550 w(28)p (#Item.45) [[381 645 392 657] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)929 b(This)551 b(is)g(ho)-36 b(w)550 b(man)-36 b(y)551 b(of)g(these)1176 9927 y(expressions)434 b(w)-36 b(ere)434 b(found.)p 52708 10518 V 800 10562 51953 45 v 800 13098 a FH(\262)421 b FL(\()p 0 1 0 0 TeXcolorcmyk(1)p (#Item.1) [[94 599 99 611] [1 1 1 [3 3]] [0 0 1]] pdfm Black 422 w(=)p 0 1 0 0 TeXcolorcmyk 420 w(2)p (#Item.2) [[112 599 122 611] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))465 b(F)-108 b(rom)463 b(the)h(pattern)f(of)i(the)f(sequence)g (in)g(item)g(1,)473 b(the)463 b(2)p FG(k)362 b FH(\241)316 b FL(1)38072 12616 y FF(th)39474 13098 y FL(term)463 b(is)i FG(k)44671 12616 y FD(2)45661 13098 y FL(and)f(the)f(2)p FG(k)51862 12616 y FF(th)800 15024 y FL(term)433 b(is)h FG(k)5936 14542 y FD(2)6757 15024 y FL(+)295 b FG(k)45 b FL(.)800 16950 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(2)p (#Item.2) [[93 564 99 576] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(use)6176 15605 y Fz(n)7186 16387 y FD(0)p FF(;)338 b FD(if)313 b FF(n)h FD(o)26 b(dd)7186 17350 y(1)p FF(;)351 b FD(if)313 b FF(n)h FD(ev)-26 b(en)12192 16950 y FL(=)13706 16317 y FD(1)p FB(\241)p FD(\()p FB(\241)p FD(1\))16842 16005 y FA(n)p 13706 16645 3703 54 v 15322 17408 a FD(2)17975 16950 y FL(for)434 b(the)f(last)h(equalit)-36 b(y)-108 b(.)800 19377 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(2)p (#Item.2) [[93 543 99 555] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(3)p (#Item.3) [[111 543 121 555] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))420 b(If)h FG(n)g FL(is)g(o)36 b(dd,)13142 18743 y FD(\()p FF(n)p FD(+1\))15647 18431 y FC(2)p 13142 19071 2967 54 v 14390 19835 a FD(4)16611 19377 y FL(=)17991 17902 y FE(j)18825 18743 y FD(\()p FF(n)p FD(+1\))21330 18431 y FC(2)p 18825 19071 V 20073 19835 a FD(4)21924 17902 y FE(k)23046 19377 y FL(since)420 b(4)h(divides)f(\()p FG(n)269 b FL(+)e(1\))35641 18894 y FD(2)36588 19377 y FL(and)419 b(if)i FG(n)g FL(is)g(ev)-36 b(en)420 b(\(=)369 b(2)p FG(k)45 b FL(\),)423 b(then)933 21245 y FD(\()p FF(n)p FD(+1\))3438 20933 y FC(2)3899 21245 y FB(\241)p FD(1)p 933 21573 4169 54 v 2782 22336 a(4)5603 21878 y FL(=)7117 21245 y FD(\(2)p FF(k)24 b FD(+1\))10035 20933 y FC(2)10496 21245 y FB(\241)p FD(1)p 7117 21573 4582 54 v 9172 22336 a(4)12200 21878 y FL(=)369 b FG(k)14302 21396 y FD(2)15123 21878 y FL(+)295 b FG(k)414 b FL(=)18900 20802 y FE(\245)19528 21878 y FG(k)20249 21396 y FD(2)21070 21878 y FL(+)294 b FG(k)341 b FL(+)24832 21355 y FD(1)p 24832 21573 471 54 v 24832 22336 a(4)25436 20802 y FE(\246)26432 21878 y FL(=)27812 20404 y FE(j)28646 21245 y FD(\(2)p FF(k)24 b FD(+1\))31564 20933 y FC(2)p 28646 21573 3380 54 v 30101 22336 a FD(4)32159 20404 y FE(k)33229 21878 y FL(=)34609 20404 y FE(j)35443 21245 y FD(\()p FF(n)p FD(+1\))37948 20933 y FC(2)p 35443 21573 2967 54 v 36691 22336 a FD(4)38542 20404 y FE(k)39243 21878 y FL(.)800 23982 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(2)p (#Item.2) [[93 501 99 513] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(4)p (#Item.4) [[111 501 121 513] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))433 b(if)g FG(n)g FL(ev)-36 b(en)432 b(\()p FG(n)370 b FL(=)e(2)p FG(k)45 b FL(\),)433 b(then)20354 22906 y FE(\245)21114 23459 y FF(n)p FD(+1)p 21114 23676 1774 54 v 21765 24440 a(2)23020 22906 y FE(\246)23869 23982 y FH(\242)24459 22906 y FE(\247)25219 23459 y FF(n)p FD(+1)p 25219 23676 V 25871 24440 a(2)27125 22906 y FE(\250)28415 23982 y FL(=)30088 22906 y FE(\245)30716 23982 y FG(k)340 b FL(+)33171 23459 y FD(1)p 33171 23676 471 54 v 33171 24440 a(2)33775 22906 y FE(\246)34623 23982 y FH(\242)35214 22906 y FE(\247)35841 23982 y FG(k)g FL(+)38297 23459 y FD(1)p 38297 23676 V 38297 24440 a(2)38900 22906 y FE(\250)40190 23982 y FL(=)661 b FG(k)45 b FL(\()p FG(k)338 b FL(+)293 b(1\))433 b(and)f(if)h FG(n)g FL(is)800 25908 y(o)36 b(dd)433 b(\(=)369 b(2)p FG(k)340 b FH(\241)296 b FL(1\),)434 b(then)13160 24832 y FE(\245)13920 25385 y FF(n)p FD(+1)p 13920 25602 1774 54 v 14572 26366 a(2)15826 24832 y FE(\246)16675 25908 y FH(\242)17265 24832 y FE(\247)18026 25385 y FF(n)p FD(+1)p 18026 25602 V 18677 26366 a(2)19932 24832 y FE(\250)21223 25908 y FL(=)664 b FG(k)23620 25426 y FD(2)24146 25908 y FL(.)800 27834 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[93 466 99 478] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))400 b(The)f(expressions)h(in)f(this)h(item)f(are)h(sho)-36 b(wn)399 b(to)h(equal)g(b)-36 b(y)400 b(using)f FH(d)37056 27311 y FF(m)p 37056 27528 833 54 v 37237 28292 a FD(2)38021 27834 y FH(e)370 b FL(=)e FH(b)41084 27311 y FF(m)p FD(+1)p 41084 27528 2035 54 v 41867 28292 a(2)43252 27834 y FH(c)p FL(,)1175 b FH(d)46101 27311 y FF(m)p 46101 27528 833 54 v 46282 28292 a FD(2)47066 27834 y FH(e)226 b(\241)g(b)49864 27311 y FF(m)p 49864 27528 V 50045 28292 a FD(2)50829 27834 y FH(c)369 b FL(=)800 28414 y Fz(n)1810 29197 y FD(1)p FF(;)338 b FD(if)313 b FF(m)h FD(o)26 b(dd)1810 30160 y(0)p FF(;)351 b FD(if)313 b FF(m)h FD(ev)-26 b(en)7078 29760 y FL(and)433 b FH(d)10330 29237 y FF(m)p FD(+1)p 10330 29454 2035 54 v 11112 30218 a(2)12497 29760 y FH(e)370 b FL(=)e FH(d)15560 29237 y FF(m)p 15560 29454 833 54 v 15741 30218 a FD(2)16526 29760 y FH(e)p FL(+)18561 28414 y Fz(n)19571 29197 y FD(0)p FF(;)338 b FD(if)313 b FF(m)h FD(o)26 b(dd)19571 30160 y(1)p FF(;)351 b FD(if)313 b FF(m)h FD(ev)-26 b(en)24839 29760 y FL(from)434 b(c)-36 b(hapter)432 b(3)i(of)h([)p 0 1 0 0 TeXcolorcmyk(20)p (#cite.gkp89) [[391 449 403 461] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)800 32186 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[93 427 99 439] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(5)p (#Item.5) [[111 427 121 439] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(item)p 0 1 0 0 TeXcolorcmyk 433 w(5)p (#Item.5) [[156 427 161 439] [1 1 1 [3 3]] [0 0 1]] pdfm Black 739 w(=)12429 31190 y FE(P)13831 31540 y FF(n)13831 32573 y(k)24 b FD(=1)15602 32186 y FH(d)16325 31663 y FF(k)p 16325 31881 514 54 v 16347 32644 a FD(2)16972 32186 y FH(e)738 b FL(=)g(2)20921 30712 y FE(\263)21715 31190 y(P)23117 31507 y FB(d)p FF(n=)p FD(2)p FB(e)23117 32556 y FF(k)24 b FD(=1)25742 32186 y FG(k)26463 30712 y FE(\264)27478 32186 y FH(\241)28945 30841 y Fz(n)29955 31623 y FB(d)p FF(n=)p FD(2)p FB(e)p FF(;)338 b FD(if)313 b FF(n)h FD(o)26 b(dd)30894 32635 y(0)p FF(;)1289 b FD(if)313 b FF(n)h FD(ev)-26 b(en)800 34688 y FL(=)738 b FH(d)3273 34165 y FF(n)p 3273 34382 571 54 v 3323 35146 a FD(2)3977 34688 y FH(e)4788 33612 y FE(\241)5397 34688 y FH(d)6120 34165 y FF(n)p 6120 34382 V 6170 35146 a FD(2)6824 34688 y FH(e)296 b FL(+)f(1)9667 33612 y FE(\242)10497 34688 y FH(\241)11964 33342 y Fz(n)12974 34125 y FB(d)p FF(n=)p FD(2)p FB(e)p FF(;)338 b FD(if)314 b FF(n)f FD(o)26 b(dd)13913 35137 y(0)p FF(;)1289 b FD(if)314 b FF(n)f FD(ev)-26 b(en)20227 34688 y FL(=)737 b FH(d)22699 34165 y FF(n)p 22699 34382 V 22749 35146 a FD(2)23403 34688 y FH(e)23993 33213 y FE(\263)24787 34688 y FH(d)25510 34165 y FF(n)p 25510 34382 V 25560 35146 a FD(2)26214 34688 y FH(e)p FL(+)28250 33342 y Fz(n)29260 34125 y FD(0)p FF(;)337 b FD(if)314 b FF(n)f FD(o)26 b(dd)29260 35088 y(1)p FF(;)350 b FD(if)314 b FF(n)f FD(ev)-26 b(en)34266 33213 y FE(\264)35428 34688 y FL(=)433 b(item)p 0 1 0 0 TeXcolorcmyk 434 w(4)p (#Item.4) [[431 405 436 417] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 36614 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[93 387 99 399] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(6)p (#Item.6) [[111 387 121 399] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Use)f FG(m)369 b FL(=)g FH(b)12439 36091 y FF(m)p 12439 36308 833 54 v 12620 37072 a FD(2)13404 36614 y FH(c)296 b FL(+)e FH(d)16319 36091 y FF(m)p 16319 36308 V 16500 37072 a FD(2)17285 36614 y FH(e)p FL(.)800 38540 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(6)p (#Item.6) [[93 370 99 382] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(In)f(the)g(last)h(line:)2751 40466 y(F)-108 b(or)433 b FG(n)370 b FL(=)e(2)p FG(m)p FL(,)800 41396 y FE(P)2202 41746 y FF(n)2202 42780 y(k)24 b FD(=)p FB(b)3999 42406 y FA(n)p FC(+2)p 3999 42578 1539 40 v 4565 43127 a(2)5670 42780 y FB(c)6365 42392 y FL(2)p FG(k)340 b FH(\241)295 b FG(n)370 b FL(=)11885 41396 y FE(P)13288 41746 y FD(2)p FF(m)13288 42780 y(k)24 b FD(=)p FF(m)p FD(+1)16844 42392 y FL(2)p FG(k)341 b FH(\241)295 b FL(2)p FG(m)369 b FL(=)23377 41396 y FE(P)24779 41746 y FF(m)24779 42780 y(i)p FD(=1)26578 42392 y FL(2)p FG(i)g FL(=)29422 41396 y FE(P)30824 41746 y FD(2)p FF(m)30824 42780 y(k)24 b FD(=)p FF(m)p FD(+1)34381 42392 y FL(2)p FG(k)340 b FH(\241)296 b FL(2)p FG(m)369 b FL(=)40913 41396 y FE(P)42315 41746 y FF(n)42315 42780 y(k)24 b FD(=)p FB(d)44112 42406 y FA(n)p FC(+1)p 44112 42578 V 44678 43127 a(2)45783 42780 y FB(e)46478 42392 y FL(2)p FG(k)340 b FH(\241)296 b FG(n)p FL(.)2751 44318 y(F)-108 b(or)433 b FG(n)370 b FL(=)e(2)p FG(m)295 b FH(\241)h FL(1,)800 45248 y FE(P)2202 45599 y FF(n)2202 46632 y(k)24 b FD(=)p FB(b)3999 46258 y FA(n)p FC(+2)p 3999 46430 V 4565 46979 a(2)5670 46632 y FB(c)6365 46244 y FL(2)p FG(k)123 b FH(\241)78 b FG(n)370 b FL(=)11451 45248 y FE(P)12853 45599 y FD(2)p FF(m)p FB(\241)p FD(1)12853 46632 y FF(k)24 b FD(=)p FF(m)15635 46244 y FL(2)p FG(k)123 b FH(\241)78 b FL(2)p FG(m)g 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26408 53492 V 26408 54602 a(8)1176 55909 y Fn(>)809 b Fm(8\(f)132 b(\))404 b Fl(rsolve\()p Fk(f)p Fl(\(n+2\)*f\(n+3\)=)5580 57716 y(\(n+3\)*f\(n+2\)+\(n+2\)*f\(n+1\)-\(n+3\)*f\(n\),)650 b(f\(2\)=2,f\(1\))640 b(=)d(1,)g(f\(0\))h(=)f(0)p Fk(g)p Fl(,f\);)11576 59731 y Fk(\241)12652 58911 y Fs(1)p 12652 59452 V 12652 60562 a(8)13593 59731 y(\()p Fk(\241)p Fs(1\))16084 59291 y FF(n)16980 59731 y Fs(+)18325 58911 y(1)p 18325 59452 V 18325 60562 a(8)19333 59731 y(+)20678 58911 y(1)p 20678 59452 V 20678 60562 a(2)21619 59731 y Fj(n)269 b Fs(+)23961 58911 y(1)p 23961 59452 V 23961 60562 a(4)24902 59731 y Fj(n)25630 59291 y FD(2)p 51909 60894 45 34119 v 800 60938 51153 45 v 800 62046 a FH(\262)445 b FL(\()p 0 1 0 0 TeXcolorcmyk(8)p (#Item.8) [[94 159 100 171] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))478 b(Using)496 b Fo(rectohomrec)473 b FL(from)478 b(the)g(Maple)g(V)g(Release)h(5)f(share)g(pac)-36 b(k)-72 b(age)478 b Fo(gfun)p FL(,)p 0 1 0 0 TeXcolorcmyk 488 w(8a)p (#Item.9) [[471 159 483 171] [1 1 1 [3 3]] [0 0 1]] pdfm Black 478 w(giv)-36 b(es)p 0 1 0 0 TeXcolorcmyk 479 w(8e)p (#Item.13) [[516 159 527 171] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 490 w(8b)p (#Item.10) [[535 159 547 171] [1 1 1 [3 3]] [0 0 1]] pdfm Black 800 63972 a(giv)g(es)p 0 1 0 0 TeXcolorcmyk 435 w(8f)p (#Item.14) [[108 141 117 153] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(and)p 0 1 0 0 TeXcolorcmyk 433 w(8c)p (#Item.11) [[144 141 155 153] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(giv)g(es)p 0 1 0 0 TeXcolorcmyk 435 w(8d)p (#Item.12) [[188 141 200 153] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 65898 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(5)p (#Item.5) [[93 124 99 136] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(9)p FG(a)p (#Item.16) [[111 124 127 136] [1 1 1 [3 3]] [0 0 1]] pdfm Black FL(\))433 b(sum)g(of)i(di\256erence,)e(see)h([)p 0 1 0 0 TeXcolorcmyk(24)p (#cite.m90) [[251 124 263 136] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)800 67424 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(7)p (#Item.7) [[93 110 99 122] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(9)p FG(a)p (#Item.16) [[111 110 127 122] [1 1 1 [3 3]] [0 0 1]] pdfm Black FL(\))433 b(the)g(generating)h(function)g(of)g(the)f(sequence)g (in)h(item)p 0 1 0 0 TeXcolorcmyk 433 w(9a)p (#Item.16) [[381 110 392 122] [1 1 1 [3 3]] [0 0 1]] pdfm Black 435 w(is)39901 66901 y FF(x)p 37471 67118 5389 54 v 37471 67882 a FD(\(1)p FB(\241)p FF(x)p FD(\)\(1)p FB(\241)p FF(x)42033 67630 y FC(2)42494 67882 y FD(\))43362 67424 y FL(=)1930 69467 y FD(1)p 933 69685 2464 54 v 933 70448 a(\(1)p FB(\241)p FF(x)p FD(\))4138 68728 y FB(1)3908 68994 y FE(P)3751 71159 y FF(k)24 b FD(=1)5688 69990 y FG(x)6427 69508 y FD(2)p FF(k)g FB(\241)p FD(1)9038 69990 y FL(=)10806 68728 y FB(1)10575 68994 y FE(P)10418 71159 y FF(k)g FD(=1)12134 69990 y FH(d)12857 69467 y FF(k)g FD(+1)p 12857 69685 1716 54 v 13480 70448 a(2)14706 69990 y FH(e)p FG(x)16035 69508 y FF(k)16604 69990 y FL(.)p Black 26150 74617 a(10)p Black eop %%Page: 11 11 11 10 bop Black 0 TeXcolorgray Black Black 800 1424 a FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(8)p (#Item.8) [[93 704 99 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(9)p (#Item.15) [[111 704 121 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Easy)g(to)g(sho)-36 b(w)p 0 1 0 0 TeXcolorcmyk 433 w(8b)p (#Item.10) [[201 704 213 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(=)p 0 1 0 0 TeXcolorcmyk(9c)p (#Item.18) [[222 704 233 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(and)p 0 1 0 0 TeXcolorcmyk 433 w(8c)p (#Item.11) [[260 704 271 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(=)p 0 1 0 0 TeXcolorcmyk(9d)p (#Item.19) [[280 704 292 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 3350 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(9)p (#Item.15) [[93 687 99 699] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(These)f(are)h(sho)-36 b(wn)434 b(to)f(b)36 b(e)434 b(equal)g(b)-36 b(y)433 b(simple)h(manipulations)g(of)g (di\256erences;)f(see)h([)p 0 1 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Fs(:=)336 b(\(1)270 b Fk(\241)f Fj(x)19300 10920 y FD(2)19825 11360 y Fs(\))202 b(\()21368 10882 y FF(@)p 21102 11080 1078 49 v 21102 11778 a(@)52 b(x)22515 11360 y Fs(F\()p Fj(x)p Fs(\)\))338 b(=)f(2)202 b(\(1)270 b(+)f(2)202 b Fj(x)p Fs(\))g(F\()p Fj(x)p Fs(\))1176 13489 y Fn(>)809 b Fl(dsolve\()p Fk(f)p Fl(ode1,F\(0\)=1)p Fk(g)p Fl(,F\(x\)\);)2837 b Fs(F\()p Fj(x)p Fs(\))337 b(=)g Fk(\241)31920 12669 y Fs(1)p 28137 13211 8174 49 v 28137 14321 a(\()p Fj(x)269 b Fs(+)g(1\))202 b(\()p Fj(x)270 b Fk(\241)f Fs(1\))35784 13971 y FD(3)1176 16077 y Fn(>)809 b Fm(10\(b\))f Fl(ode2:=\(1-x)13489 15637 y FD(2)14019 16077 y Fl(\)*diff\(F\(x\),x\)=1+\(4+3*x-2*x)31191 15637 y FD(2)31726 16077 y Fl(+x)32998 15637 y FD(3)33525 16077 y Fl(\)*F\(x\);)11576 17884 y Fi(o)-62 b(de2)500 b Fs(:=)336 b(\(1)270 b Fk(\241)f Fj(x)19300 17444 y FD(2)19825 17884 y Fs(\))202 b(\()21368 17407 y FF(@)p 21102 17605 1078 49 v 21102 18302 a(@)52 b(x)22515 17884 y Fs(F\()p Fj(x)p Fs(\)\))338 b(=)f(1)269 b(+)g(\(4)h(+)f(3)202 b Fj(x)269 b Fk(\241)g Fs(2)202 b Fj(x)36159 17444 y FD(2)36954 17884 y Fs(+)270 b Fj(x)38860 17444 y FD(3)39385 17884 y Fs(\))202 b(F\()p Fj(x)p Fs(\))1176 20023 y Fn(>)809 b Fl(simplify\(dsolve\()p Fk(f)p Fl(ode2,F\(0\)=0)p Fk(g)p Fl(,F\(x\)\)\);)2841 b Fs(F\()p Fj(x)p Fs(\))337 b(=)f Fk(\241)38241 19203 y Fj(x)p 34500 19744 8174 49 v 34500 20854 a Fs(\()p Fj(x)270 b Fs(+)f(1\))202 b(\()p Fj(x)270 b Fk(\241)f Fs(1\))42148 20504 y FD(3)1176 22610 y Fn(>)539 b Fm(10\(c\))g Fl(ode3:=\(1-x)12796 22170 y FD(2)13325 22610 y Fl(\)*diff\(F\(x\),x,x\)=\(4+5*x-2*x)30497 22170 y FD(2)31032 22610 y Fl(+x)32304 22170 y FD(3)32831 22610 y Fl(\)*diff\(F\(x\),x\)+\(3-4*x+3*x)48731 22170 y FD(2)49266 22610 y Fl(\)*F\(x\);)7576 24417 y Fi(o)-62 b(de3)500 b Fs(:=)336 b(\(1)104 b Fk(\241)g Fj(x)14969 23977 y FD(2)15494 24417 y Fs(\))202 b(\()17036 23940 y FF(@)17584 23627 y FC(2)p 16771 24138 1539 49 v 16771 24867 a FF(@)52 b(x)17849 24615 y FC(2)18645 24417 y Fs(F\()p Fj(x)p Fs(\)\))338 b(=)e(\(4)104 b(+)g(5)202 b Fj(x)104 b Fk(\241)g Fs(2)202 b Fj(x)29540 23977 y FD(2)30168 24417 y Fs(+)104 b Fj(x)31908 23977 y FD(3)32433 24417 y Fs(\))202 b(\()33975 23940 y FF(@)p 33710 24138 1078 49 v 33710 24835 a(@)52 b(x)35123 24417 y Fs(F\()p Fj(x)p Fs(\)\))104 b(+)g(\(3)g Fk(\241)g Fs(4)202 b Fj(x)104 b Fs(+)g(3)202 b Fj(x)45552 23977 y FD(2)46076 24417 y Fs(\))g(F\()p Fj(x)p Fs(\))1176 26556 y Fn(>)809 b Fl(dsolve\()p Fk(f)p Fl(ode3,F\(0\)=0,D\(F\)\(0\)=1) p Fk(g)p Fl(,F\(x\)\);)2841 b Fs(F\()p Fj(x)p Fs(\))337 b(=)f Fk(\241)38241 25736 y Fj(x)p 34500 26277 8174 49 v 34500 27387 a Fs(\()p Fj(x)270 b Fs(+)f(1\))202 b(\()p Fj(x)270 b Fk(\241)f Fs(1\))42148 27037 y FD(3)1176 29143 y Fn(>)809 b Fm(10\(d\))f Fl(ode4:=\(1-x)13489 28704 y FD(2)14019 29143 y Fl(\)*diff\(F\(x\),x\)=2x+\(6+2*x-4*x)31827 28704 y FD(2)32362 29143 y Fl(+2*x)34906 28704 y FD(3)35434 29143 y Fl(\)*F\(x\);)7576 30950 y Fi(o)-62 b(de4)500 b Fs(:=)336 b(\(1)270 b Fk(\241)f Fj(x)15300 30510 y FD(2)15825 30950 y Fs(\))202 b(\()17368 30473 y FF(@)17916 30160 y FC(2)p 17103 30671 1539 49 v 17103 31400 a FF(@)52 b(x)18181 31148 y FC(2)18977 30950 y Fs(F\()p Fj(x)p Fs(\)\))337 b(=)g(2)p Fj(x)269 b Fs(+)g(\(6)h(+)f(2)202 b Fj(x)269 b Fk(\241)g Fs(4)202 b Fj(x)33313 30510 y FD(2)34108 30950 y Fs(+)270 b(2)202 b Fj(x)36822 30510 y FD(3)37347 30950 y Fs(\))g(F\()p Fj(x)p Fs(\))1176 33341 y Fn(>)809 b Fl(dsolve\()p Fk(f)p Fl(ode4,F\(0\)=0)p Fk(g)p Fl(,F\(x\)\);)2837 b Fs(F\()p Fj(x)p Fs(\))337 b(=)g Fk(\241)31614 32521 y Fj(x)32307 32081 y FD(2)p 28137 33062 8174 49 v 28137 34173 a Fs(\()p Fj(x)269 b Fs(+)g(1\))202 b(\()p Fj(x)270 b Fk(\241)f Fs(1\))35784 33822 y FD(3)1176 35929 y Fn(>)539 b Fm(10\(e\))g Fl(ode5:=x*\(1-x) 14087 35489 y FD(2)14617 35929 y Fl(\)*diff\(F\(x\),x,x\)=\(1+6*x+3*x) 31789 35489 y FD(2)32325 35929 y Fl(-4*x)34869 35489 y FD(3)35396 35929 y Fl(+2x)37304 35489 y FD(4)37831 35929 y Fl(\)*F\(x\)+\(-6-4x)46099 35489 y FD(2)46629 35929 y Fl(+4x)48537 35489 y FD(3)49064 35929 y Fl(\)*F\(x\);)2776 37735 y Fi(o)-62 b(de5)500 b Fs(:=)336 b Fj(x)p Fs(\(1)57 b Fk(\241)g Fj(x)10768 37296 y FD(2)11294 37735 y Fs(\))202 b(\()12836 37258 y FF(@)13384 36946 y FC(2)p 12572 37456 1539 49 v 12572 38185 a FF(@)52 b(x)13650 37933 y FC(2)14445 37735 y Fs(F\()p Fj(x)p Fs(\)\))338 b(=)f(\(1)57 b(+)g(6)202 b Fj(x)57 b Fs(+)g(3)202 b Fj(x)25153 37296 y FD(2)25737 37735 y Fk(\241)57 b Fs(4)202 b Fj(x)28238 37296 y FD(3)28821 37735 y Fs(+)57 b(2)202 b Fj(x)31322 37296 y FD(4)31848 37735 y Fs(\)\()33188 37258 y FF(@)p 32923 37456 1078 49 v 32923 38153 a(@)52 b(x)34336 37735 y Fs(F\()p Fj(x)p Fs(\)\))57 b(+)g(\()p Fk(\241)p Fs(6)g Fk(\241)g Fs(4)p Fj(x)42666 37296 y FD(2)43251 37735 y Fs(+)g(4)p Fj(x)45550 37296 y FD(3)46076 37735 y Fs(\))202 b(F\()p Fj(x)p Fs(\))1176 40127 y Fn(>)809 b Fl(dsolve\()p Fk(f)p Fl (ode5,F\(0\)=0,D\(D\(F\)\)\(0\)=2)p Fk(g)p Fl(,F\(x\)\);)2842 b Fs(F\()p Fj(x)p Fs(\))337 b(=)g Fk(\241)39887 39307 y Fj(x)40580 38867 y FD(2)p 36410 39848 8174 49 v 36410 40958 a Fs(\()p Fj(x)269 b Fs(+)g(1\))202 b(\()p Fj(x)270 b Fk(\241)f Fs(1\))44057 40608 y FD(3)p 49509 41593 45 33383 v 800 41637 48753 45 v 800 42744 a FH(\262)485 b FL(\()p 0 1 0 0 TeXcolorcmyk(1)p (#Item.1) [[94 332 100 344] [1 1 1 [3 3]] [0 0 1]] pdfm Black 485 w(=)p 0 1 0 0 TeXcolorcmyk 485 w(10)p (#Item.22) [[113 332 129 344] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))502 b Fo(listtodiffeq)496 b FL(from)502 b(Maple)g(V)f(R5)h (share)g(pac)-36 b(k)-72 b(age)502 b Fo(gfun)e FL(w)-36 b(as)502 b(used)f(to)h(get)p 0 1 0 0 TeXcolorcmyk 502 w(10a)p (#Item.23) [[504 332 521 344] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 520 w(10b)p (#Item.24) [[529 332 547 344] [1 1 1 [3 3]] [0 0 1]] pdfm Black 800 44671 a(and)p 0 1 0 0 TeXcolorcmyk 433 w(10d)p (#Item.26) [[102 315 120 327] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 46597 y FH(\262)535 b FL(\()p 0 1 0 0 TeXcolorcmyk(10)p (#Item.22) [[95 298 106 310] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))c(Using)550 b Fo(diffeqtohomdiffeq)523 b FL(from)531 b(Maple)g(V)g(Release)h(5)f(share)g(pac)-36 b(k)-72 b(age)532 b Fo(gfun)p FL(,)p 0 1 0 0 TeXcolorcmyk 554 w(10b)p (#Item.24) [[499 298 518 310] [1 1 1 [3 3]] [0 0 1]] pdfm Black 531 w(giv)-36 b(es)p 0 1 0 0 TeXcolorcmyk 800 48523 a(10c)p (#Item.25) [[79 280 96 292] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(and)p 0 1 0 0 TeXcolorcmyk 433 w(10d)p (#Item.26) [[123 280 141 292] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(giv)g(es)p 0 1 0 0 TeXcolorcmyk 435 w(10e)p (#Item.27) [[174 280 190 292] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 50449 y FH(\262)559 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[95 263 101 275] [1 1 1 [3 3]] [0 0 1]] pdfm Black 559 w(=)p 0 1 0 0 TeXcolorcmyk 559 w(11)p (#Item.28) [[115 263 131 275] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))545 b(A)g(quadratic)h FG(f)142 b FL(\()p FG(x)p FL(\))558 b(=)h FG(ax)21194 49967 y FD(2)22090 50449 y FL(+)371 b FG(bx)g FL(+)g FG(c)545 b FL(with)g(in)-36 b(teger)545 b(co)36 b(e\261cien)-36 b(ts)546 b(and)e FG(a)h FL(negativ)-36 b(e)546 b(has)800 52375 y(its)568 b(maxim)-36 b(um)568 b(v)-72 b(alue)569 b(at)f FG(x)598 b FL(=)f FH(b)17840 51852 y FB(\241)p FF(b)p 17840 52069 1135 54 v 17922 52833 a FD(2)p FF(a)19107 52375 y FH(c)569 b FL(and)e FG(x)598 b FL(=)f FH(d)26598 51852 y FB(\241)p FF(b)p 26598 52069 V 26680 52833 a FD(2)p FF(a)27866 52375 y FH(e)p FL(.)981 b(So)568 b(item)p 0 1 0 0 TeXcolorcmyk 568 w(11)p (#Item.28) [[386 246 397 258] [1 1 1 [3 3]] [0 0 1]] pdfm Black 569 w(=)1156 b(Max)38314 53260 y FF(k)24 b FB(2f)p FD(1)p FF(::n)p FB(g)42181 52375 y FH(\241)p FG(k)43935 51893 y FD(2)44848 52375 y FL(+)386 b(\()p FG(n)h FL(+)g(1\))p FG(k)642 b FL(=)800 55119 y(\()p FG(n)356 b FL(+)g(1)h FH(\241)f(b)6925 54596 y FF(n)p FD(+1)p 6925 54814 1774 54 v 7576 55577 a(2)8831 55119 y FH(c)p FL(\))p FH(b)10650 54596 y FF(n)p FD(+1)p 10650 54814 V 11302 55577 a(2)12557 55119 y FH(c)521 b FL(=)g FH(d)15924 54596 y FF(n)p FD(+1)p 15924 54814 V 16576 55577 a(2)17830 55119 y FH(eb)19143 54596 y FF(n)p FD(+1)p 19143 54814 V 19795 55577 a(2)21050 55119 y FH(c)h FL(=)g(item)p 0 1 0 0 TeXcolorcmyk 523 w(4)p (#Item.4) [[313 221 319 233] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)546 b(since)524 b FG(m)355 b FH(\241)i(b)35188 54596 y FF(m)p 35188 54814 833 54 v 35369 55577 a FD(2)36153 55119 y FH(c)522 b FL(=)f FH(b)39521 54596 y FF(m)p 39521 54814 V 39702 55577 a FD(2)40486 55119 y FH(c)p FL(+)42611 53774 y Fz(n)43621 54556 y FD(1)p FF(;)337 b FD(if)314 b FF(n)g FD(o)26 b(dd)43621 55519 y(0)p FF(;)351 b FD(if)313 b FF(n)h FD(ev)-26 b(en)48627 55119 y FL(=)521 b FH(d)50883 54596 y FF(m)p 50883 54814 V 51064 55577 a FD(2)51848 55119 y FH(e)p FL(.)800 57045 y(Similarly)435 b(for)f FG(x)369 b FL(=)g FH(d)11527 56522 y FF(n)p FD(+1)p 11527 56740 1774 54 v 12178 57504 a(2)13433 57045 y FH(e)p FL(.)800 58972 y FH(\262)g FL(\()p 0 1 0 0 TeXcolorcmyk(11)p (#Item.28) [[93 186 105 198] [1 1 1 [3 3]] [0 0 1]] pdfm Black 370 w(=)p 0 1 0 0 TeXcolorcmyk 368 w(12)p (#Item.29) [[117 186 132 198] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Since)f(item)p 0 1 0 0 TeXcolorcmyk 434 w(12)p (#Item.29) [[198 186 210 198] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(=)1805 b(Max)17103 59857 y Fu(A)p FB(2)p FD(P)-26 b(art\(1)p FF(::n)p FD(\))22724 58972 y FH(j)p Ft(A)p FH(j)296 b(\242)f FL(Max)25643 59807 y FF(A)p FB(2)p Fu(A)28125 58972 y FH(j)p FG(A)p FH(j)369 b FL(=)1086 b(Max)31587 59857 y FF(m)p FB(2f)p FD(1)p FF(::n)p FB(g)35773 58972 y FG(m)2048 b FL(Max)37132 59857 y Fu(A)p FB(2)p FD(P)-26 b(art)40236 59968 y FA(m)41018 59857 y FD(\(1)p FF(::n)p FD(\))43535 58972 y FL(Max)43804 59807 y FF(A)p FB(2)p Fu(A)46285 58972 y FH(j)p FG(A)p FH(j)369 b FL(=)1518 61318 y(Max)800 62204 y FF(m)p FB(2f)p FD(1)p FF(::n)p FB(g)4986 61318 y FG(m)p FL(\()p FG(n)311 b FH(\241)g FG(m)g FL(+)g(1\))408 b(=)457 b(item)p 0 1 0 0 TeXcolorcmyk 457 w(11)p (#Item.28) [[233 165 244 177] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)463 b(where)457 b(P)-36 b(art)26268 61517 y FF(m)27156 61318 y FL(\(1)p FG(::n)p FL(\))457 b(are)g(the)g(set)g(partitions)f (of)i FH(f)p FL(1)p FG(::n)p FH(g)g FL(with)f FG(m)800 63591 y FL(blo)36 b(c)-36 b(ks.)800 65517 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(13)p (#Item.30) [[93 127 105 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black 370 w(=)p 0 1 0 0 TeXcolorcmyk 368 w(15)p (#Item.32) [[117 127 132 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))427 b(The)g(Robinson-Sc)-36 b(hensted-Kn)g(uth)423 b(algorithm)k([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.c94) [[364 127 370 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)h(p.218],)g([)p 0 1 0 0 TeXcolorcmyk(35)p (#cite.sw86) [[418 127 429 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)g(p.94])g(giv)-36 b(es)428 b(a)f(bijection)800 67443 y(b)36 b(et)-36 b(w)g(een)501 b(p)36 b(erm)-36 b(utations)501 b(of)h FH(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)p FH(g)503 b FL(and)e(ordered)f(pairs)i(of)g(Y)-108 b(oung)501 b(tableaux)h(of)g FG(n)g FL(of)g(the)f(same)800 69370 y(shap)36 b(e,)633 b(where)593 b(the)f(n)-36 b(um)g(b)36 b(er)591 b(of)j(ro)-36 b(ws)593 b(of)h(the)e(tableaux)i(is)f(the)g (length)f(of)i(the)e(longest)i(increasing)800 71296 y(subsequence)301 b(of)i(the)f(p)36 b(erm)-36 b(utation)301 b(and)h(the)f(n)-36 b(um)g(b)36 b(er)301 b(of)i(columns)f(is)g(length)g(of)h(the)e(longest) i(decreasing)p Black 26150 74617 a(11)p Black eop %%Page: 12 12 12 11 bop Black 0 TeXcolorgray Black Black 800 1424 a FL(subsequence.)800 3682 y(The)502 b FK(RSK)g FL(algorithm)h(as)f(used) g(in)g(C.)h(C.)f(Rousseau's)h FJ(Partitions)527 b(and)h(q-series)f(in)g (c)-66 b(ombinatorics)800 5608 y FL(course)434 b(at)f(the)g(Univ)-36 b(ersit)g(y)434 b(of)h(Memphis)e(in)g(spring)g(2000.)p 800 6550 44753 45 v 800 39836 45 33286 v 1176 7712 a Fh(Algorithm)426 b(3.1:)492 b Fg(RSK)p Fv(\()p Ff(n;)184 b Fe(h)p Ff(a)14146 7878 y Fd(i)14517 7712 y Fe(i)14947 7311 y Fd(n)14947 8000 y(i)p Fc(=1)16437 7712 y Fv(\))1653 10180 y(INPUT:)370 b(n,)g(a)f(p)31 b(ositiv)-31 b(e)371 b(in)-31 b(teger)1653 11774 y(INPUT:)370 b(\()p Ff(a)6957 11940 y Fd(i)7327 11774 y Fv(\))7757 11372 y Fd(n)7757 12062 y(i)p Fc(=1)9246 11774 y Fv(,)g(a)f(p)31 b(erm)-31 b(utation)372 b(of)d Fe(f)p Fv(1)p Ff(::n)p Fe(g)1653 13368 y Fv(OUTPUT:)i(\()p Ff(P)31 b(;)184 b(Q)p Fv(\),)371 b(a)f(pair)f(of)h(standard)f(Y)-92 b(oung)370 b(tableaux)h(of)f(order)f Ff(n)5973 14962 y Fv(and)g(b)31 b(oth)370 b(of)g(the)f(same)h(shap)31 b(e)1653 18150 y Ff(P)154 b Fv([)184 b Ff(;)g Fv(])309 b(:=)f Fe(;)p Ff(;)184 b(Q)p Fv([)g Ff(;)g Fv(])312 b(:=)c Fe(;)1107 b Fh(commen)-35 b(t:)238 b Fv(these)369 b(are)g(empt)-31 b(y)371 b(2D)e(arra)-31 b(ys)1653 20221 y Fh(for)425 b Ff(p)308 b Fv(:=)f(1)370 b Fh(to)425 b Ff(n)2499 28784 y Fh(do)4451 20869 y Fb(8)4451 21866 y(>)4451 22198 y(>)4451 22530 y(>)4451 22862 y(>)4451 23194 y(>)4451 23526 y(>)4451 23858 y(>)4451 24190 y(>)4451 24522 y(>)4451 24854 y(>)4451 25186 y(>)4451 25519 y(>)4451 25851 y(>)4451 26183 y(>)4451 26515 y(>)4451 26847 y(>)4451 27179 y(>)4451 27511 y(<)4451 29504 y(>)4451 29836 y(>)4451 30168 y(>)4451 30500 y(>)4451 30832 y(>)4451 31164 y(>)4451 31496 y(>)4451 31828 y(>)4451 32160 y(>)4451 32493 y(>)4451 32825 y(>)4451 33157 y(>)4451 33489 y(>)4451 33821 y(>)4451 34153 y(>)4451 34485 y(>)4451 34817 y(>)4451 35149 y(:)5435 21815 y Ff(b)308 b Fv(:=)g Ff(a)8279 21981 y Fd(p)5435 23409 y Ff(r)339 b Fv(:=)308 b(1)5435 25003 y Fh(while)425 b Fv(ro)-31 b(w)369 b(r)g(is)g(not)h (empt)-31 b(y)386 b Fh(and)369 b Ff(b)g Fv(is)g(not)h(greater)g(than)g (the)f(last)h(cell)g(in)g(ro)-31 b(w)369 b Ff(r)400 b Fv(of)370 b Ff(P)6281 28149 y Fh(do)8233 25547 y Fb(8)8233 26544 y(>)8233 26876 y(<)8233 28868 y(>)8233 29200 y(:)9217 26597 y Ff(c)308 b Fv(:=)g(Min)p Fe(f)p Ff(j)371 b Fe(j)308 b Ff(b)g Fe(\267)f Ff(P)154 b Fv(\()p Ff(r)-31 b(;)184 b(j)63 b Fv(\))p Fe(g)9217 28191 y Fv(sw)-31 b(ap)q(\()p Ff(b;)184 b(P)154 b Fv(\()p Ff(r)-31 b(;)184 b(c)p Fv(\)\))9217 29785 y Ff(r)339 b Fv(:=)308 b Ff(r)277 b Fv(+)246 b(1)5435 31379 y Fh(commen)-35 b(t:)238 b Fv(add)370 b(a)f(new)h(cell)g(at)g (end)e(of)i(ro)-31 b(w)370 b Ff(r)400 b Fv(of)370 b Ff(P)522 b Fv(and)370 b Ff(Q)5435 32650 y(c)308 b Fv(:=)g(1)246 b(+)g(the)369 b(n)-31 b(um)g(b)31 b(er)370 b(of)f(cells)h(in)f(ro)-31 b(w)370 b(r)5435 34244 y Ff(P)154 b Fv(\()p Ff(r)-31 b(;)184 b(c)p Fv(\))310 b(:=)e Ff(b)5435 35838 y(Q)p Fv(\()p Ff(r)-31 b(;)184 b(c)p Fv(\))310 b(:=)e Ff(p)1653 39026 y Fh(return)426 b Fv(\()p Ff(P)31 b(;)184 b(Q)p Fv(\))p 45508 39836 V 800 39880 44753 45 v 45531 40323 465 33375 v 1243 40323 44753 465 v 2751 42377 a FL(F)-108 b(or)462 b(a)g(partition)g(of)h FG(n)p FL(,)469 b FG(a)p FL(,)h(the)461 b(#ro)-36 b(ws\(shap)36 b(eRSK\()p FG(n;)221 b(a)p FL(\))418 b(=)462 b(the)f(size)i(of)f(longest)h(increasing)f (sub-)800 44304 y(sequence)569 b(of)g FG(a)g FL(and)f(#cols\(shap)36 b(eRSK)q(\()p FG(n;)221 b(a)p FL(\))599 b(=)569 b(the)f(size)h(of)h (longest)f(decreasing)g(subsequence)f(of)800 46230 y FG(a)p FL(.)p Black 26150 74617 a(12)p Black eop %%Page: 13 13 13 12 bop Black 0 TeXcolorgray Black Black 800 1424 a FL(The)434 b(in)-36 b(v)g(erse)433 b(of)i(the)e FK(RSK)g FL(algorithm.)p 800 2366 44753 45 v 800 31976 45 29610 v 1176 3528 a Fh(Algorithm)426 b(3.2:)492 b Fg(iRSK)q Fv(\()p Ff(n;)184 b Fe(h)p Ff(P)31 b(;)184 b(Q)p Fe(i)p Fv(\))1653 5984 y(INPUT:)370 b(n,)g(a)f(p)31 b(ositiv)-31 b(e)371 b(in)-31 b(teger)1653 7578 y(INPUT:)370 b(\()p Ff(P)31 b(;)184 b(Q)p Fv(\),)371 b(a)f(pair)f(of)h(standard)g(Y)-92 b(oung)370 b(tableaux)h(of)e(order)g Ff(n)5973 9172 y Fv(and)g(b)31 b(oth)370 b(of)g(the)f(same)h(shap)31 b(e)1653 10766 y(OUTPUT:)371 b(\()p Ff(a)8218 10932 y Fd(i)8587 10766 y Fv(\))9017 10365 y Fd(n)9017 11055 y(i)p Fc(=1)10507 10766 y Fv(,)f(a)f(p)31 b(erm)-31 b(utation)371 b(of)f Fe(f)p Fv(1)p Ff(::n)p Fe(g)1653 13955 y Fh(for)425 b Ff(p)308 b Fv(:=)f Ff(n)370 b Fh(do)-35 b(wn)g(to)425 b Fv(1)2499 21721 y Fh(do)4451 14470 y Fb(8)4451 15466 y(>)4451 15799 y(>)4451 16131 y(>)4451 16463 y(>)4451 16795 y(>)4451 17127 y(>)4451 17459 y(>)4451 17791 y(>)4451 18123 y(>)4451 18455 y(>)4451 18787 y(>)4451 19119 y(>)4451 19452 y(>)4451 19784 y(>)4451 20116 y(>)4451 20448 y(<)4451 22440 y(>)4451 22772 y(>)4451 23105 y(>)4451 23437 y(>)4451 23769 y(>)4451 24101 y(>)4451 24433 y(>)4451 24765 y(>)4451 25097 y(>)4451 25429 y(>)4451 25761 y(>)4451 26093 y(>)4451 26425 y(>)4451 26758 y(>)4451 27090 y(>)4451 27422 y(:)5435 15549 y Fv(\()p Ff(r)-31 b(;)184 b(c)p Fv(\))310 b(:=)e(\257nd)369 b(the)h(ro)-31 b(w)369 b(and)h(column)g(of)g(the)f(v)-61 b(alue)370 b(of)g Ff(p)f Fv(in)g(arra)-31 b(y)370 b Ff(Q)5435 17143 y(b)308 b Fv(:=)g Ff(P)154 b Fv(\()p Ff(r)-31 b(;)184 b(c)p Fv(\))5435 18737 y(delete)370 b(cell)g(\()p Ff(r)-31 b(;)184 b(c)p Fv(\))372 b(of)d Ff(P)5435 23315 y Fh(while)425 b Ff(r)338 b Fe(6)p Fv(=)308 b(1)369 b Fh(do)13658 19385 y Fb(8)13658 20381 y(>)13658 20713 y(>)13658 21046 y(>)13658 21378 y(>)13658 21710 y(>)13658 22042 y(<)13658 24034 y(>)13658 24366 y(>)13658 24699 y(>)13658 25031 y(>)13658 25363 y(>)13658 25695 y(:)14642 20331 y Ff(r)338 b Fv(:=)308 b Ff(r)278 b Fe(\241)246 b Fv(1)14642 21925 y Fh(commen)-35 b(t:)238 b Fv(in)369 b(ro)-31 b(w)370 b Ff(r)400 b Fv(of)370 b Ff(P)523 b Fv(put)369 b Ff(b)g Fv(in)g(the)h(correct)f(sp)31 b(ot)20242 23195 y(and)369 b(pass)g(bac)-31 b(k)370 b(the)f(bump)31 b(ed)369 b(v)-61 b(alue)370 b(as)f Ff(b)14642 24789 y(c)307 b Fv(:=)h(Max)q Fe(f)p Ff(j)371 b Fe(j)308 b Ff(P)154 b Fv(\()p Ff(r)-31 b(;)184 b(j)63 b Fv(\))310 b Ff(<)d(b)p Fe(g)14642 26383 y Fv(sw)-31 b(ap)q(\()p Ff(b;)184 b(P)154 b Fv(\()p Ff(r)-31 b(;)184 b(c)p Fv(\)\))5435 27977 y Ff(a)6020 28143 y Fd(p)6839 27977 y Fv(:=)308 b Ff(b)1653 31165 y Fh(return)426 b Fv(\(\()p Ff(a)7064 31331 y Fd(i)7433 31165 y Fv(\))7863 30764 y Fd(n)7863 31453 y(i)p Fc(=1)9353 31165 y Fv(\))p 45508 31976 V 800 32020 44753 45 v 45531 32463 465 29699 v 1243 32463 44753 465 v 2751 34517 a FL(F)-108 b(or)289 b FG(P)36 b(;)221 b(Q)290 b FL(StdY)-108 b(oungT)g(ab)287 b(of)j FG(n)g FL(with)f(the)f(same)i(shap)36 b(e,)318 b(then)288 b(iRSK\()p FG(n;)221 b FL(\()p FG(P)36 b(;)221 b(Q)p FL(\)\))41840 34035 y FB(\241)p FD(1)43468 34517 y FL(=)369 b(iRSK\()p FG(n;)221 b FL(\()p FG(Q;)g(P)181 b FL(\)\))800 37107 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(12)p (#Item.29) [[93 383 105 395] [1 1 1 [3 3]] [0 0 1]] pdfm Black 370 w(=)p 0 1 0 0 TeXcolorcmyk 368 w(14)p (#Item.31) [[117 383 132 395] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Use)g(fact)p 0 1 0 0 TeXcolorcmyk 434 w(2.7)p (#fact.2.7) [[186 383 201 395] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 39034 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(14)p (#Item.31) [[93 366 105 378] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(use)f(fact)p 0 1 0 0 TeXcolorcmyk 435 w(2.7)p (#fact.2.7) [[157 366 171 378] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(to)h(sho)-36 b(w)433 b(that)h(comp)36 b(ositions)434 b(and)f(partitions)g(of)i FG(n)f FL(giv)-36 b(e)434 b(the)f(same)h (result.)800 40960 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[93 348 99 360] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(14)p (#Item.31) [[111 348 126 360] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(The)g(partitions)g(\()p FH(b)16623 40437 y FF(n)p FD(+1)p 16623 40654 1774 54 v 17274 41418 a(2)18529 40960 y FH(c)p FG(;)221 b FL(1)p FG(;)g(:)g(:)g(:)k(;)c FL(1)19701 41709 y FE(|)p 20299 41709 912 160 v 912 w({z)p 22407 41709 V 912 w(})19771 43064 y FB(d)20322 42691 y FA(n)p Fx(\241)p FC(1)p 20322 42863 1557 40 v 20898 43412 a(2)22012 43064 y FB(e)314 b FD(1's)23916 40960 y FL(\))370 b(and)f(\()p FH(d)28486 40437 y FF(n)p FD(+1)p 28486 40654 1774 54 v 29137 41418 a(2)30392 40960 y FH(e)p FG(;)221 b FL(1)p FG(;)g(:)g(:)g(:)k(;)c FL(1)31564 41709 y FE(|)p 32162 41709 912 160 v 912 w({z)p 34270 41709 V 912 w(})31635 43064 y FB(b)32186 42691 y FA(n)p Fx(\241)p FC(1)p 32186 42863 1557 40 v 32761 43412 a(2)33875 43064 y FB(c)314 b FD(1's)35779 40960 y FL(\))370 b(are)g(\(the)f(only\))h (partitions)g(of)h FG(n)800 44749 y FL(whic)-36 b(h)398 b(ac)-36 b(hiev)g(e)398 b(the)g(maxim)-36 b(um)398 b(v)-72 b(alue)398 b(since)h FH(b)24404 44226 y FF(n)p FD(+1)p 24404 44443 1774 54 v 25055 45207 a(2)26310 44749 y FH(c)223 b FL(+)f FH(d)29080 44226 y FF(n)p FB(\241)p FD(1)p 29080 44443 V 29731 45207 a(2)30986 44749 y FH(e)369 b FL(=)g FG(n)398 b FL(and)g FH(d)37717 44226 y FF(n)p FD(+1)p 37717 44443 V 38368 45207 a(2)39623 44749 y FH(e)223 b FL(+)f FH(b)42393 44226 y FF(n)p FB(\241)p FD(1)p 42393 44443 V 43045 45207 a(2)44299 44749 y FH(c)370 b FL(=)e FG(n)399 b FL(and)e(they)800 46675 y(are)502 b(equal)h(if)g FG(n)g FL(is)f(o)36 b(dd.)784 b(But)501 b(for)i(the)f(\257rst)f (partition,)519 b(max)p FH(\242)p FL(len)487 b(=)e FH(b)36519 46152 y FF(n)p FD(+1)p 36519 46369 V 37170 47133 a(2)38425 46675 y FH(c)342 b(\242)40068 45599 y FE(\241)40677 46675 y FH(d)41400 46152 y FF(n)p FB(\241)p FD(1)p 41400 46369 V 42052 47133 a(2)43306 46675 y FH(e)296 b FL(+)f(1)46149 45599 y FE(\242)47244 46675 y FL(=)501 b(item)p 0 1 0 0 TeXcolorcmyk 502 w(4)p (#Item.4) [[538 297 544 309] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)800 48601 y(and)433 b(for)h(the)f(second)h(max)p FH(\242)p FL(len)369 b(=)g FH(d)18679 48078 y FF(n)p FD(+1)p 18679 48296 V 19330 49059 a(2)20585 48601 y FH(e)296 b(\242)22135 47525 y FE(\241)22744 48601 y FH(b)23467 48078 y FF(n)p FB(\241)p FD(1)p 23467 48296 V 24118 49059 a(2)25373 48601 y FH(c)f FL(+)g(1)28215 47525 y FE(\242)29193 48601 y FL(=)434 b(item)p 0 1 0 0 TeXcolorcmyk 433 w(4)p (#Item.4) [[374 280 380 292] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 50527 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(14)p (#Item.31) [[93 262 105 274] [1 1 1 [3 3]] [0 0 1]] pdfm Black 370 w(=)p 0 1 0 0 TeXcolorcmyk 368 w(15)p (#Item.32) [[117 262 132 274] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Use)g(fact)p 0 1 0 0 TeXcolorcmyk 434 w(2.7)p (#fact.2.7) [[186 262 201 274] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 52453 y FH(\262)391 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[93 245 99 257] [1 1 1 [3 3]] [0 0 1]] pdfm Black 391 w(=)p 0 1 0 0 TeXcolorcmyk 390 w(16)p (#Item.33) [[112 245 127 257] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))447 b(It)f(is)h(kno)-36 b(wn)446 b(that)g FG(\302)p FL(\()p FG(G)p FL(\))303 b(+)h FG(\302)p FL(\()p 22554 51386 1026 54 v FG(G)o FL(\))391 b FH(\267)f FG(n)305 b FL(+)e(1)447 b(for)g(an)-36 b(y)446 b(graph)g FG(G)g FL(with)h FG(n)f FL(v)-36 b(ertices)447 b([)p 0 1 0 0 TeXcolorcmyk(26)p (#cite.ng56) [[507 245 518 257] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)g([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.cl96) [[532 245 544 257] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)800 54379 y(p.)1214 b(232].)i(No)-36 b(w)646 b(if)g FG(G)730 b FL(=)g FG(K)16169 54723 y FB(d)16720 54350 y FA(n)p FC(+1)p 16719 54522 1539 40 v 17285 55071 a(2)18390 54723 y FB(e)19303 54379 y FH(])440 b FL(\()p FG(n)f FH(\241)h(d)24546 53856 y FF(n)p FD(+1)p 24546 54074 1774 54 v 25198 54837 a(2)26452 54379 y FH(e)p FL(\))p FG(K)28655 54578 y FD(1)29181 54379 y FL(,)699 b(then)644 b FG(\302)p FL(\()p FG(G)p FL(\))730 b(=)f FG(\302)p FL(\()p FG(K)41167 54723 y FB(d)41718 54350 y FA(n)p FC(+1)p 41718 54522 1539 40 v 42284 55071 a(2)43389 54723 y FB(e)43862 54379 y FL(\))h(=)f FH(d)47562 53856 y FF(n)p FD(+1)p 47562 54074 1774 54 v 48214 54837 a(2)49469 54379 y FH(e)646 b FL(and)800 56305 y FG(\302)p FL(\()p 2121 55238 1026 54 v FG(G)p FL(\))368 b(=)h FG(\302)p FL(\()p FG(K)7830 56504 y FF(n)8751 56305 y FH(\241)295 b FG(K)11186 56649 y FB(d)11737 56276 y FA(n)p FC(+1)p 11737 56448 1539 40 v 12303 56997 a(2)13408 56649 y FB(e)13881 56305 y FL(\))369 b(=)g FG(n)295 b FL(+)g(1)h FH(\241)f(d)21512 55782 y FF(n)p FD(+1)p 21512 56000 1774 54 v 22164 56764 a(2)23418 56305 y FH(e)370 b FL(=)e FH(b)26481 55782 y FF(n)p FD(+1)p 26481 56000 V 27133 56764 a(2)28387 56305 y FH(c)p FL(.)579 b(No)-36 b(w)434 b(prop)36 b(osition)p 0 1 0 0 TeXcolorcmyk 434 w(2.6)p (#fact.2.6) [[429 210 444 222] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 58232 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(3)p (#Item.3) [[93 193 99 205] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(17)p (#Item.34) [[111 193 126 205] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))386 b(Let)g FG(G)369 b FL(=)f(\()p FG(n)198 b FH(\241)g(d)15423 57709 y FF(n)p 15423 57926 571 54 v 15473 58690 a FD(2)16127 58232 y FH(e)p FL(\))p FH(\242)p FL(K)18603 58431 y FD(1)19327 58232 y FH(])g FL(K)21421 58438 y FB(d)21972 58076 y FA(n)p 21972 58237 512 40 v 22025 58785 a FC(2)22616 58438 y FB(e)23090 58232 y FL(,)395 b(then)385 b FG(!)48 b FL(\()p FG(G)p FL(\))368 b(=)h FH(d)32129 57709 y FF(n)p 32129 57926 571 54 v 32179 58690 a FD(2)32833 58232 y FH(e)386 b FL(and,)395 b(since)386 b FG(n)369 b FL(=)g FH(d)43049 57709 y FF(n)p 43049 57926 V 43099 58690 a FD(2)43753 58232 y FH(e)198 b FL(+)g FH(b)46474 57709 y FF(n)p 46474 57926 V 46524 58690 a FD(2)47177 58232 y FH(c)p FL(,)p 48524 57500 858 54 v 396 w FG(!)48 b FL(\()p FG(G)p FL(\))368 b(=)800 60290 y FH(b)1523 59767 y FF(n)p 1523 59985 571 54 v 1573 60748 a FD(2)2227 60290 y FH(c)169 b FL(+)g(1,)386 b(so)p 7099 59559 858 54 v 372 w FG(!)48 b FL(\()p FG(G)p FL(\))169 b FH(\241)g FG(!)48 b FL(\()p FG(G)p FL(\))369 b(=)f(1)169 b FH(\241)g FL(\()p FH(d)19261 59767 y FF(n)p 19261 59985 571 54 v 19311 60748 a FD(2)19967 60290 y FH(e)g(\241)g(b)22651 59767 y FF(n)p 22651 59985 V 22701 60748 a FD(2)23356 60290 y FH(c)p FL(\))369 b(=)26205 58945 y Fz(n)27215 59727 y FD(0)p FF(;)337 b FD(if)314 b FF(n)f FD(o)26 b(dd)27215 60690 y(1)p FF(;)350 b FD(if)314 b FF(n)f FD(ev)-26 b(en)32221 60290 y FL(.)558 b(W)-108 b(e)372 b(also)h(ha)-36 b(v)g(e)p 40802 59559 858 54 v 372 w FG(!)48 b FL(\()p FG(H)103 b FL(\))169 b(+)g FG(!)48 b FL(\()p FG(H)103 b FL(\))367 b FH(\267)i FG(n)169 b FL(+)g(1)800 62216 y(for)434 b(ev)-36 b(ery)434 b FG(H)472 b FH(2)368 b FL(graph\()p FG(n)p FL(\),)434 b(so)g(use)f(prop)36 b(osition)p 0 1 0 0 TeXcolorcmyk 434 w(2.6)p (#fact.2.6) [[302 157 317 169] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 64142 y FH(\262)458 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[94 140 100 152] [1 1 1 [3 3]] [0 0 1]] pdfm Black 458 w(=)p 0 1 0 0 TeXcolorcmyk 457 w(18)p (#Item.35) [[113 140 129 152] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))487 b(It)e(is)h(kno)-36 b(wn)486 b(that)g(1)331 b(+)f(\242\()p FG(G)p FL(\))g(+)h FG(\260)72 b FL(\()p 25531 63075 1026 54 v FG(G)o FL(\))458 b FH(\267)g FG(n)331 b FL(+)f(1)486 b(for)h(an)-36 b(y)486 b(graph)f FG(G)h FL(with)g FG(n)g FL(v)-36 b(ertices)486 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.b73) [[538 140 544 152] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)800 66068 y(p.)850 b(304].)i(Let)524 b FG(G)f FL(=)g FH(d)12486 65545 y FF(n)p FB(\241)p FD(1)p 12486 65763 1774 54 v 13138 66527 a(2)14393 66068 y FH(e\242)p FL(K)16363 66267 y FD(1)17246 66068 y FH(])356 b FL(K)19500 66412 y FD(1)p FF(;)p FB(b)20782 66039 y FA(n)p Fx(\241)p FC(1)p 20782 66211 1557 40 v 21358 66760 a(2)22472 66412 y FB(c)22945 66068 y FL(,)548 b(then)523 b(1)357 b(+)g(\242\()p FG(G)p FL(\))523 b(=)g(1)357 b(+)g FH(b)37562 65545 y FF(n)p FB(\241)p FD(1)p 37562 65763 1774 54 v 38213 66527 a(2)39468 66068 y FH(c)524 b FL(=)f FH(b)42840 65545 y FF(n)p FD(+1)p 42840 65763 V 43491 66527 a(2)44746 66068 y FH(c)i FL(and)f FG(\260)72 b FL(\()p FG(G)p FL(\))522 b(=)800 67995 y(1)296 b(+)e FH(d)3775 67472 y FF(n)p FB(\241)p FD(1)p 3775 67689 V 4427 68453 a(2)5682 67995 y FH(e)369 b FL(=)g FH(d)8745 67472 y FF(n)p FD(+1)p 8745 67689 V 9396 68453 a(2)10651 67995 y FH(e)p FL(.)579 b(note)433 b(that)g FH(j)p FG(V)289 b FL(\()p FG(G)p FL(\))p FH(j)369 b FL(=)g FH(d)24185 67472 y FF(n)p FB(\241)p FD(1)p 24185 67689 V 24836 68453 a(2)26091 67995 y FH(e)295 b FL(+)g FH(b)29006 67472 y FF(n)p FB(\241)p FD(1)p 29006 67689 V 29658 68453 a(2)30912 67995 y FH(c)h FL(+)f(1)369 b(=)g FG(n)p FL(.)800 69921 y FH(\262)g FL(\()p 0 1 0 0 TeXcolorcmyk(3)p (#Item.3) [[93 88 99 100] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(19)p (#Item.36) [[111 88 126 100] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(See)f(prop)36 b(osition)p 0 1 0 0 TeXcolorcmyk 434 w(2.6)p (#fact.2.6) [[217 88 232 100] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)p Black 26150 74617 a(13)p Black eop %%Page: 14 14 14 13 bop Black 0 TeXcolorgray Black Black 800 1424 a FH(\262)381 b FL(\()p 0 1 0 0 TeXcolorcmyk(5)p (#Item.5) [[93 704 99 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black 380 w(=)p 0 1 0 0 TeXcolorcmyk 381 w(20)p (#Item.37) [[112 704 127 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))440 b(The)h(n)-36 b(um)g(b)36 b(er)438 b(of)j(graphs)g(with)f (only)h FG(m)f FL(lo)36 b(ops)441 b(on)f(t)-36 b(w)g(o)441 b(v)-36 b(ertices)441 b(is)f(equal)h(to)g(the)f(n)-36 b(um)g(b)36 b(er)800 3350 y(of)375 b(partitions)g(of)g FG(m)f FL(with)h(at)g(most)f(t)-36 b(w)g(o)375 b(parts)f(\(=)369 b FH(b)26872 2827 y FF(m)p FD(+2)p 26872 3045 2035 54 v 27654 3808 a(2)29040 3350 y FH(c)p FL(\).)558 b(Of)375 b(the)f FG(n)175 b FH(\241)g FL(1)375 b(edges)g(if)g FG(k)414 b FH(2)369 b(f)p FL(1)p FG(;)221 b(:)g(:)g(:)j(;)d(n)175 b FH(\241)g FL(1)p FH(g)800 5276 y FL(are)438 b(b)36 b(et)-36 b(w)g(een)437 b(v)-36 b(ertices,)440 b(there)d(are)h(then)f FH(b)22259 4753 y FF(n)p FB(\241)p FD(1)p FB(\241)p FF(k)24 b FD(+2)p 22259 4971 4221 54 v 24135 5734 a(2)26613 5276 y FH(c)439 b FL(graphs)e(with)h(the)f(remaining)h(edges.)592 b(Hence)438 b(the)800 7202 y(total)c(n)-36 b(um)g(b)36 b(er)432 b(of)i(graphs)g(is)15628 6206 y FE(P)17030 6557 y FF(n)p FB(\241)p FD(1)17030 7590 y FF(k)24 b FD(=0)18858 7202 y FH(b)19581 6679 y FF(n)p FB(\241)p FD(1)p FB(\241)p FF(k)g FD(+2)p 19581 6897 V 21457 7660 a(2)23935 7202 y FH(c)370 b FL(=)26275 6206 y FE(P)27677 6557 y FF(n)p FB(\241)p FD(1)27677 7590 y FF(k)24 b FD(=0)29506 7202 y FH(b)30229 6679 y FF(k)g FD(+2)p 30229 6897 1716 54 v 30852 7660 a(2)32078 7202 y FH(c)434 b FL(whic)-36 b(h)433 b(is)h(item)p 0 1 0 0 TeXcolorcmyk 434 w(5)p (#Item.5) [[442 652 448 664] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 9129 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(6)p (#Item.6) [[93 635 99 647] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(22)p (#Item.39) [[111 635 126 647] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))392 b(F)-108 b(rom)390 b(the)h(follo)-36 b(wing)393 b(table)e(of)h(the)f(triangles)h(with)f(largest)h(side)f FG(n)p FL(,)400 b(w)-36 b(e)392 b(see)f(that)g(the)f(total)800 11997 y(n)-36 b(um)g(b)36 b(er)288 b(of)h(triangles)h(is)13026 10506 y FB(b)13577 10132 y FA(n)p Fx(\241)p FC(1)p 13577 10304 1557 40 v 14153 10853 a(2)15267 10506 y FB(c)13655 11000 y FE(P)13498 13165 y FF(k)24 b FD(=0)15685 11997 y FL(\()p FG(n)p FH(\241)p FL(2)p FG(k)45 b FL(\))290 b(whic)-36 b(h)288 b(is)i(item)p 0 1 0 0 TeXcolorcmyk 289 w(6)p (#Item.6) [[322 609 327 621] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)2495 b(n)p 32596 12575 45 1927 v 1328 w(sides)434 b(of)g(triangle)p 30567 12619 25997 45 v 31231 13967 a(1)p 32596 14545 45 1927 v 1401 w(111)31231 15893 y(2)p 32596 16471 V 1401 w(222)h(221)31231 17819 y(3)p 32596 18397 V 1401 w(333)g(332)f(331)h(,)f(322)31231 19745 y(4)p 32596 20323 V 1401 w(444)h(443)f(442)h(441)f(,)g(433)h(432)31231 21672 y(5)p 32596 22249 V 1401 w(555)g(554)f(553)h(552)f(551)h(,)f(544) g(543)h(542)f(,)g(533)2751 23283 y(Note)g(the)f(strict)g(triangular)h (inequalit)-36 b(y)435 b(will)g(b)36 b(e)433 b(satis\257ed)g(for)h(in) -36 b(teger)434 b(sided)f(triangles.)800 25209 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(1)p (#Item.1) [[93 490 99 502] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(22)p (#Item.39) [[111 490 126 502] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(See)f([)p 0 1 0 0 TeXcolorcmyk(22)p (#cite.jm00amm) [[159 490 171 502] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)800 27135 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(9)p FG(c)p (#Item.18) [[93 473 104 485] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w FL(=)p 0 1 0 0 TeXcolorcmyk 369 w(23)p (#Item.40) [[116 473 131 485] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Let)f FG(k)414 b FL(=)369 b(1)433 b(in)p 0 1 0 0 TeXcolorcmyk 434 w(23)p (#Item.40) [[206 473 218 485] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)h(see)g([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.a85) [[248 473 253 485] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)800 29061 y FH(\262)504 b FL(\()p 0 1 0 0 TeXcolorcmyk(9)p FG(a)p (#Item.16) [[94 455 106 467] [1 1 1 [3 3]] [0 0 1]] pdfm Black 504 w FL(=)p 0 1 0 0 TeXcolorcmyk 504 w(24)p (#Item.41) [[120 455 136 467] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))513 b(F)-108 b(rom)513 b(the)f(de\257nition)g(of)i(the)e (Losanitsc)-36 b(h)513 b(n)-36 b(um)g(b)36 b(er)511 b(follo)-36 b(wing)515 b(the)d(table)h(of)h(v)-72 b(alues)513 b(of)800 30987 y FG(L)p FL(\()p FG(r)-36 b(;)221 b(c)p FL(\),)570 b(w)-36 b(e)542 b(ha)-36 b(v)g(e)543 b FG(L)p FL(\(3)p FG(;)221 b(c)369 b FL(+)g(1\))h FH(\241)f FG(L)p FL(\(3)p FG(;)221 b(c)p FL(\))554 b(=)g FG(L)p FL(\(2)p FG(;)221 b(c)369 b FL(+)g(1\))554 b(=)g(1)p FG(;)221 b FL(2)p FG(;)g FL(2)p FG(;)g FL(3)p FG(;)g FL(3)p FG(;)g FL(4)p FG(;)g FL(4)p FG(;)g(:)g(:)g(:)772 b FL(and)542 b FG(L)p FL(\(2)p FG(;)221 b FL(1\))555 b(=)e(1,)800 32913 y(whic)-36 b(h)433 b(is)h(item)p 0 1 0 0 TeXcolorcmyk 434 w(9a)p (#Item.16) [[151 421 163 433] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 34839 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(25)p (#Item.42) [[93 403 105 415] [1 1 1 [3 3]] [0 0 1]] pdfm Black 370 w(=)p 0 1 0 0 TeXcolorcmyk 368 w(26)p (#Item.43) [[117 403 132 415] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))313 b FG(a)8191 35038 y FF(n;k)9961 34839 y FL(=)11342 33365 y FE(n)17130 34193 y FD(1)p FF(;)5043 b FD(if)314 b FF(k)285 b FD(=)261 b(1)12448 35215 y FB(jf)p FF(U)94 b FB(2)p FF(A)15242 35326 y FA(n)15810 35215 y FB(j)p FD(min\()p FF(U)g FD(\)=)p FF(k)24 b FB(\241)p FD(1)p FB(gj)p FF(;)362 b FD(if)314 b FF(k)285 b FB(6)p FD(=)261 b(1)26359 34839 y FL(,)337 b(where)312 b FG(a)31376 35038 y FF(n;k)33090 34839 y FL(is)g(the)g(v)-72 b(alues)313 b(of)g(the)f(arra)-36 b(y)313 b(in)f(item)p 0 1 0 0 TeXcolorcmyk 312 w(26)p (#Item.43) [[532 403 544 415] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)800 36765 y(and)433 b FG(A)4304 36964 y FF(n)5364 36765 y FL(is)h(as)g(in)f(item)p 0 1 0 0 TeXcolorcmyk 434 w(25)p (#Item.42) [[187 386 198 398] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)579 b(\(this)433 b(is)h(ho)-36 b(w)434 b(the)f(arra)-36 b(y)434 b(in)f(item)p 0 1 0 0 TeXcolorcmyk 434 w(26)p (#Item.43) [[361 386 373 398] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(w)-36 b(as)434 b(found\))800 38691 y FH(\262)610 b FL(\()p 0 1 0 0 TeXcolorcmyk(2)p (#Item.2) [[95 369 101 381] [1 1 1 [3 3]] [0 0 1]] pdfm Black 609 w(=)p 0 1 0 0 TeXcolorcmyk 609 w(26)p (#Item.43) [[116 369 133 381] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))575 b(If)h FG(n)f FL(is)g(ev)-36 b(en)575 b(item)p 0 1 0 0 TeXcolorcmyk 575 w(26)p (#Item.43) [[237 369 248 381] [1 1 1 [3 3]] [0 0 1]] pdfm Black 575 w(=)609 b(2)22658 37695 y FE(P)24194 37524 y FA(n)p 24194 37685 512 40 v 24247 38234 a FC(2)24061 39061 y FF(k)24 b FD(=1)26054 38691 y FG(k)654 b FL(=)29138 38168 y FF(n)p 29138 38386 571 54 v 29188 39149 a FD(2)29842 38691 y FL(\()30481 38168 y FF(n)p 30481 38386 V 30531 39149 a FD(2)31575 38691 y FL(+)391 b(1\))610 b(=)574 b(item)p 0 1 0 0 TeXcolorcmyk 575 w(2)p (#Item.2) [[427 369 433 381] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)1002 b(If)576 b FG(n)f FL(is)g(o)36 b(dd)575 b(item)p 0 1 0 0 TeXcolorcmyk 574 w(26)p (#Item.43) [[535 369 547 381] [1 1 1 [3 3]] [0 0 1]] pdfm Black 800 40866 a(=)369 b(2)3052 39870 y FE(P)4587 39688 y FA(n)p Fx(\241)p FC(1)p 4587 39860 1557 40 v 5163 40409 a(2)4455 41236 y FF(k)24 b FD(=1)6554 40866 y FG(k)340 b FL(+)9009 40343 y FF(n)p FD(+1)p 9009 40561 1774 54 v 9661 41324 a(2)11284 40866 y FL(=)12798 40343 y FF(n)p FB(\241)p FD(1)p 12798 40561 V 13449 41324 a(2)14704 40866 y FL(\()15343 40343 y FF(n)p FB(\241)p FD(1)p 15343 40561 V 15994 41324 a(2)17544 40866 y FL(+)295 b(1\))g(+)21742 40343 y FF(n)p FD(+1)p 21742 40561 V 22393 41324 a(2)24017 40866 y FL(=)433 b(item)p 0 1 0 0 TeXcolorcmyk 433 w(2)p (#Item.2) [[328 349 334 361] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 42793 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(2)p (#Item.2) [[93 332 99 344] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(27)p (#Item.44) [[111 332 126 344] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Let)f FG(n)369 b FL(=)g(2)p FG(k)479 b FL(and)433 b(=)369 b(2)p FG(k)340 b FH(\241)296 b FL(1.)578 b(See)434 b(c)-36 b(hapter)432 b(6)i(of)h([)p 0 1 0 0 TeXcolorcmyk(29)p (#cite.r78) [[360 332 372 344] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])e(for)h(partitions.)800 44719 y FH(\262)415 b FL(\()p 0 1 0 0 TeXcolorcmyk(28)p (#Item.45) [[93 315 105 327] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))460 b(use:)631 b(if)461 b FG(n)415 b FL(=)f(2)p FG(k)506 b FL(then)459 b FH(b)16837 44196 y FF(n)p FD(+1)p 16837 44413 V 17489 45177 a(2)18743 44719 y FH(c)415 b FL(=)f FH(d)21897 44196 y FF(n)p 21897 44413 571 54 v 21947 45177 a FD(2)22601 44719 y FH(e)h FL(=)f FG(k)45 b FL(,)467 b FH(b)27304 44196 y FF(n)p FD(+2)p 27304 44413 1774 54 v 27955 45177 a(2)29210 44719 y FH(c)415 b FL(=)f FH(d)32364 44196 y FF(n)p FD(+1)p 32364 44413 V 33015 45177 a(2)34270 44719 y FH(e)h FL(=)f FG(k)358 b FL(+)313 b(1,)468 b(and)459 b FH(b)43817 44196 y FF(n)p FD(+3)p 43817 44413 V 44469 45177 a(2)45724 44719 y FH(c)414 b FL(=)g FH(d)48877 44196 y FF(n)p FD(+2)p 48877 44413 V 49529 45177 a(2)50784 44719 y FH(e)g FL(=)800 46645 y FG(k)340 b FL(+)295 b(1.)2751 48571 y(if)311 b FG(n)370 b FL(=)e(2)p FG(k)88 b FL(+)44 b(1)311 b(then)f FH(b)13340 48048 y FF(n)p FD(+1)p 13340 48265 V 13991 49029 a(2)15246 48571 y FH(c)369 b FL(=)g FH(d)18309 48048 y FF(n)p 18309 48265 571 54 v 18359 49029 a FD(2)19013 48571 y FH(e)g FL(=)g FG(k)88 b FL(+)44 b(1,)335 b FH(b)25242 48048 y FF(n)p FD(+2)p 25242 48265 1774 54 v 25894 49029 a(2)27148 48571 y FH(c)370 b FL(=)e FH(d)30211 48048 y FF(n)p FD(+1)p 30211 48265 V 30863 49029 a(2)32118 48571 y FH(e)h FL(=)g FG(k)88 b FL(+)44 b(1,)335 b(and)310 b FH(b)40753 48048 y FF(n)p FD(+3)p 40753 48265 V 41405 49029 a(2)42659 48571 y FH(c)370 b FL(=)e FH(d)45722 48048 y FF(n)p FD(+2)p 45722 48265 V 46374 49029 a(2)47629 48571 y FH(e)h FL(=)g FG(k)88 b FL(+)44 b(2.)800 50497 y FH(\262)369 b FL(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Item.4) [[93 263 99 275] [1 1 1 [3 3]] [0 0 1]] pdfm Black 369 w(=)p 0 1 0 0 TeXcolorcmyk 369 w(28)p (#Item.45) [[111 263 126 275] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))434 b(Let)f FG(s)369 b FL(=)f(3)434 b(and)f FG(m)369 b FL(=)g FG(n)295 b FL(+)g(1)434 b(in)f(T)-108 b(ur\266)-650 b(an's)434 b(theorem.)2751 52423 y(Ev)-36 b(ery)374 b(graph)e(on)h(m)g(v)-36 b(ertices)373 b(not)g(con)-36 b(taining)373 b(a)g(complete)g(graph)g(of)g FG(s)g FL(v)-36 b(ertices,)386 b FG(K)45023 52622 y FF(s)45513 52423 y FL(,)f(has)373 b(at)g(most)800 54349 y FG(ex)p FL(\()p FG(m)p FL(;)221 b FG(K)5570 53670 y FD(\(2\))5475 54485 y FF(s)6828 54349 y FL(\))433 b(v)-36 b(ertices.)p Black 800 56623 a FK(Prop)42 b(osition)500 b(3.1)e(\(T)-125 b(ur)-10 b(\266)-737 b(an[)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.a95amm) [[218 207 224 219] [1 1 1 [3 3]] [0 0 1]] pdfm Black 3 w(,)p 0 1 0 0 TeXcolorcmyk 498 w(25)p (#cite.ms65cjm) [[233 207 246 219] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\))p Black 650 w FJ(L)-66 b(et)509 b FL(2)452 b FH(\267)g FG(m;)221 b(s)510 b FJ(b)-66 b(e)509 b(p)-66 b(ositive)508 b(inte)-66 b(gers,)519 b(then)509 b(the)g(fol)66 b(lowing)510 b(ar)-66 b(e)800 58549 y(e)g(qual.)p Black 2338 62171 a(1.)p Black 4052 61095 a FE(\241)4661 61582 y FF(m)4841 62629 y FD(2)5493 61095 y FE(\242)6397 62171 y FH(\241)7866 60511 y FF(s)p FB(\241)p FD(2)7725 60909 y FE(X)7923 63709 y FF(i)p FD(=0)9865 60298 y FE(\265)10843 61273 y FH(b)11566 60750 y FF(m)p FD(+)p FF(i)p 11566 60967 1885 54 v 11690 61731 a(s)p FB(\241)p FD(1)13584 61273 y FH(c)12183 63082 y FL(2)14174 60298 y FE(\266)15152 62171 y FJ(,)1793 b(se)-66 b(e)563 b FL([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.b78) [[251 157 257 169] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)434 b(p.294],[)p 0 1 0 0 TeXcolorcmyk(7)p (#cite.b86) [[301 157 307 169] [1 1 1 [3 3]] [0 0 1]] pdfm Black 2 w(,)g(p.54])p Black 2338 66913 a FJ(2.)p Black 5620 65651 a FE(X)4052 68457 y FD(0)p FB(\267)p FF(i)g(n;)221 b(n)297 b FL(+)e FG(k)45 b FL(\))1697 b(=)46005 47932 y FE(\265)46983 48906 y FG(n)295 b FH(\241)h FL(1)48183 50716 y(2)50033 47932 y FE(\266)800 53940 y FJ(wher)-66 b(e)417 b FG(p)5075 53458 y FB(\244)5075 54268 y FD(2)5600 53940 y FL(\()p FG(m)p FL(\))368 b(=)417 b FJ(the)f(numb)-66 b(er)416 b(of)h(p)-66 b(artitions)415 b(of)h FG(m)h FJ(with)g(two)498 b FL(distinct)416 b FJ(p)-66 b(arts,)426 b(and)417 b FG(p)43215 53458 y FB(\244)43215 54268 y FD(2)43741 53940 y FL(\(max)369 b FG(>)g(n;)221 b(m)p FL(\))369 b(=)800 55866 y FJ(the)414 b(numb)-66 b(er)413 b(of)h(p)-66 b(artitions)412 b(of)i FG(m)g FJ(with)g(two)496 b FL(distinct)413 b FJ(p)-66 b(arts,)424 b(the)414 b(lar)-66 b(gest)413 b(p)-66 b(art)414 b(gr)-66 b(e)g(ater)412 b(than)i FG(n)p FJ(.)581 b(Se)-66 b(e)512 b FL([)p 0 1 0 0 TeXcolorcmyk(3)p (#cite.a94) [[538 214 544 226] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)800 57792 y(Ch.12,13,14])p FJ(,)p FL([)p 0 1 0 0 TeXcolorcmyk(28)p (#cite.p56) [[149 197 161 209] [1 1 1 [3 3]] [0 0 1]] pdfm Black 4 w(],[)p 0 1 0 0 TeXcolorcmyk(29)p (#cite.r78) [[171 197 182 209] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)434 b(Ch.6])466 b FJ(for)e(p)-66 b(artitions.)800 62185 y Fr(4)1793 b(Ac)-50 b(kno)g(wledgemen)g(ts)800 65427 y FL(Thanks)515 b(to)g(the)f(referee)h(for)h(suggestions,)535 b(and)515 b(ap)36 b(ologies)516 b(to)f(the)f(editor)h(for)g(m)-36 b(y)515 b(dela)-36 b(y)516 b(in)e(making)800 67353 y(the)433 b(c)-36 b(hanges.)p Black 26150 74617 a(15)p Black eop %%Page: 16 16 16 15 bop Black 0 TeXcolorgray Black Black 800 1424 a Fr(References)p Black 1406 3644 a Fs([1])p Black 650 w(M.)404 b(Aigner,)f(T)-101 b(ur\266)-606 b(an's)405 b(graph)g(theorem,)f Fi(A)-31 b(mer.)433 b(Math.)f(Monthly)p Fs(,)402 b Fm(102)i Fs(\(1995\),)h(808{816.)p 0 1 0 0 TeXcolorcmyk 941 w(14)p (#page.14) [[493 684 504 696] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 5899 a([2])p Black 650 w(G.)363 b(L.)f(Alexanderson)h (et)h(al.,)e Fi(The)395 b(Wil)62 b(liam)395 b(Powel)62 b(l)395 b(Putnam)g(Mathematic)-62 b(al)393 b(Comp)-62 b(etition)394 b(-)h(Pr)-62 b(oblems)3336 7368 y(and)427 b(Solutions:)e(1965-1984)p Fs(,)392 b(Mathematical)397 b(Asso)34 b(ciation)397 b(of)h(America,,)d(1985.)i(\(Problem)g(A-1)g (of)h(27)51863 6928 y FF(th)3336 8691 y Fs(Comp)34 b(etition\))p 0 1 0 0 TeXcolorcmyk 943 w(14)p (#page.14) [[174 639 185 651] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 10946 a([3])p Black 650 w(G.)404 b(E.)f(Andrews,)i Fi(Numb)-62 b(er)433 b(The)-62 b(ory)p Fs(,)402 b(Do)-34 b(v)g(er,)404 b(1994.)g(Corrected)g(reprin)-34 b(t)404 b(of)h(1971)f(edition.)p 0 1 0 0 TeXcolorcmyk 941 w(15)p (#page.15) [[492 618 503 630] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 13201 a([4])p Black 650 w(E.)430 b(J.)h(Barb)34 b(eau,)430 b(M.)h(S.)g(Klamkin,)f(and)i(W.)f(O.)f(J.)h(Moser,)f Fi(Five)458 b(Hundr)-62 b(e)g(d)457 b(Mathematic)-62 b(al)455 b(Chal)62 b(lenges)p Fs(,)3336 14456 y(Mathematical)404 b(Asso)34 b(ciation)404 b(of)h(America,)e(1995.)p 0 1 0 0 TeXcolorcmyk 941 w(6)p (#page.6) [[322 587 327 599] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(15)p (#page.15) [[334 587 345 599] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 16644 a([5])p Black 650 w(C.)h(Berge,)f Fi(Gr)-62 b(aphs)431 b(and)j(Hyp)-62 b(er)g(gr)g(aphs)p Fs(,)400 b(North)405 b(Holland,)f(1973.)p 0 1 0 0 TeXcolorcmyk 941 w(13)p (#page.13) [[383 567 394 579] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 18899 a([6])p Black 650 w(B.)f(Bollob\266)-606 b(as,)404 b Fi(Extr)-62 b(emal)433 b(Gr)-62 b(aph)432 b(The)-62 b(ory)p Fs(,)402 b(Academic)i(Press,)f(1978.)p 0 1 0 0 TeXcolorcmyk 942 w(14)p (#page.14) [[398 547 409 559] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 21154 a([7])p Black 650 w(B.)g(Bollob\266)-606 b(as,)404 b Fi(Combinatorics)p Fs(,)d(Cam)-34 b(bridge)405 b(Univ)-34 b(ersit)g(y)404 b(Press,)f(1986.)p 0 1 0 0 TeXcolorcmyk 942 w(14)p (#page.14) [[412 527 423 539] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 23410 a([8])p Black 650 w(P)-101 b(.)404 b(J.)f(Cameron,)h Fi(Combinatorics)p Fs(,)e(Cam)-34 b(bridge)405 b(Univ)-34 b(ersit)g(y)404 b(Press,)f(1994.)p 0 1 0 0 TeXcolorcmyk 942 w(4)p (#page.4) [[426 506 431 518] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 403 w(11)p (#page.11) [[438 506 449 518] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 1406 25665 a([9])p Black 650 w(P)-101 b(.)391 b(J.)f(Cameron,)i(Sequences)f(realized)e(b)-34 b(y)392 b(oligomorphic)f(p)34 b(erm)-34 b(utation)392 b(groups,)g(Article)e (00.1.5,)g(V)-101 b(ol.)391 b Fm(3)3336 26920 y Fi(J.)434 b(Inte)-62 b(ger)433 b(Se)-62 b(q.)p Fs(,)403 b(2000,)h(published)h (electronically)d(at)p 0 1 0 0 TeXcolorcmyk 3336 28176 a Fi(http://www.r)-62 b(ese)g(ar)g(ch.att.c)g(om/~njas/se)g(quenc)g (es/JIS/V)-31 b(OL3/gr)-62 b(oups.html)p [[102 463 434 475] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~{}njas/sequences/JIS/VOL3/groups.html) pdfm Black Fs(.)p 0 1 0 0 TeXcolorcmyk 926 w(6)p (#page.6) [[445 463 451 475] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 30431 a([10])p Black 650 w(G.)404 b(Chartrand)h(and)g (L.)f(Lesniak,)f Fi(Gr)-62 b(aphs)431 b(&)j(Digr)-62 b(aphs)p Fs(,)402 b(3ed,)i(Chapman)i(&)e(Hall,)f(1996.)p 0 1 0 0 TeXcolorcmyk 941 w(1)p (#page.1) [[491 443 497 455] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(13)p (#page.13) [[503 443 514 455] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 32686 a([11])p Black 650 w(E.)798 b(J.)f(Co)34 b(c)-34 b(k)-67 b(a)-34 b(yne,)798 b(D.)g(McCrea)g(and)g(C.)h(M.)e (Mynhardt,)j(Nordhaus-Gaddum)h(result)d(for)g(CO-)3336 34009 y(irredundance)404 b(in)h(graphs)f Fi(Disc.)433 b(Math.)402 b Fm(211)i Fs(\(2000\),)h(209{215.)p 0 1 0 0 TeXcolorcmyk 941 w(8)p (#page.8) [[384 411 390 423] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 36264 a([12])p Black 650 w(E.)507 b(J.)h(Co)34 b(c)-34 b(k)-67 b(a)-34 b(yne)508 b(and)h(C.)f(M.)f(Mynhardt,)i(On)f (the)h(pro)34 b(duct)509 b(of)f(upp)34 b(er)508 b(irredundance)g(n)-34 b(um)g(b)34 b(ers)510 b(of)e(a)3336 37587 y(graph)405 b(and)g(its)f(complemen)-34 b(t,)404 b Fi(Disc.)433 b(Math.)402 b Fm(76)i Fs(\(1989\),)h(117{121.)p 0 1 0 0 TeXcolorcmyk 941 w(8)p (#page.8) [[396 379 401 391] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 39842 a([13])p Black 650 w(R.)553 b(Diestel,)g Fi(Gr)-62 b(aph)570 b(The)-62 b(ory)p Fs(,)552 b(Springer,)i(1997.)f (\(Second)i(edition,)f(2000\))g(Av)-67 b(ailable)552 b(at)p 0 1 0 0 TeXcolorcmyk 554 w Fi(Gr)-62 b(aph)570 b(The-)p [[491 358 547 370] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uni-hamburg.de/home/diestel/) pdfm 3336 41165 a(ory,2e)-62 b(d)p [[79 347 136 359] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uni-hamburg.de/home/diestel/) pdfm Black Fs(,)401 b(www.math.uni-ham)-34 b(burg.de/home/diestel/)p 0 1 0 0 TeXcolorcmyk 945 w(1)p (#page.1) [[353 347 359 359] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 43420 a([14])p Black 650 w(Encyclop)34 b(edia)403 b(of)i(Com)-34 b(binatorial)405 b(Structures,)g(h)-34 b(ttp://algo.inria.fr/bin/encyclop)34 b(edia.)p 0 1 0 0 TeXcolorcmyk 941 w(6)p (#page.6) [[496 326 502 338] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 45675 a([15])p Black 650 w(H.)501 b(J.)g(Finc)-34 b(k,)501 b(On)g(the)h(c)-34 b(hromatic)501 b(n)-34 b(um)g(b)34 b(er)502 b(of)g(a)f(graph)h(and)g(its)f(complemen)-34 b(t,)502 b(in)f(P)-101 b(.)501 b(Erd\304)-606 b(os)501 b(and)h(G.)3336 46930 y(Katona,)319 b(eds.,)g Fi(The)-62 b(ory)354 b(of)h(Gr)-62 b(aphs,)353 b(Pr)-62 b(o)g(c)g(e)g(e)g(dings) 353 b(of)i(the)f(Col)62 b(lo)-62 b(quium)353 b(held)g(at)i(Tihany,)f (Hungary,)g(1966)p Fs(,)3336 48118 y(Academic)403 b(Press,)h(1968,)g (pp.)g(99{113.)p 0 1 0 0 TeXcolorcmyk 942 w(6)p (#page.6) [[275 284 281 296] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 50306 a([16])p Black 650 w(E.)379 b(Fix)f(and)i(J.)f (L.)f(Ho)34 b(dges,)379 b(Jr.,)f(Signi\257cance)i(probabilities)f(of)g (the)h(Wilco)-34 b(xon)378 b(test,)i Fi(A)-31 b(nn.)411 b(Math.)e(Stat.)3336 51629 y Fm(26)404 b Fs(\(1955\),)h(301{312.)p 0 1 0 0 TeXcolorcmyk 941 w(6)p (#page.6) [[205 252 210 264] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 53884 a([17])p Black 650 w(W.)475 b(F)-101 b(ulton,)476 b Fi(Y)-93 b(oung)498 b(T)-93 b(able)-62 b(aux)p Fs(,)473 b(London)j(Mathematics)g(So)34 b(ciet)-34 b(y)475 b(Studen)-34 b(t)478 b(T)-101 b(ext)475 b(V)-101 b(ol.)475 b(35,)g(Cam)-34 b(bridge)3336 55126 y(Univ)g(ersit)g(y)404 b(Press,)f(1997.)p 0 1 0 0 TeXcolorcmyk 942 w(4)p (#page.4) [[220 221 225 233] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 57314 a([18])p Black 650 w(S.)353 b(Getu,)g(L.)e(W.)h (Shapiro,)h(W.-J.)f(W)-101 b(oan,)353 b(and)g(L.)f(C.)h(W)-101 b(o)34 b(o)g(dson,)352 b(Ho)-34 b(w)354 b(to)f(guess)g(a)f(generating)h (function,)3336 58637 y Fi(SIAM)434 b(J.)g(Disc.)e(Math.)403 b Fm(5)h Fs(\(1992\),)h(497{499.)p 0 1 0 0 TeXcolorcmyk 941 w(6)p (#page.6) [[304 189 309 201] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 60892 a([19])p Black 650 w(R.)592 b(L.)f(Graham,)h(M.)g (Gr\304)-606 b(otsc)-34 b(hel)592 b(and)g(L.)f(Lo)-34 b(v)-67 b(asz,)591 b(eds.,)h Fi(Handb)-62 b(o)g(ok)604 b(of)i(Combinatorics,)e(V)-93 b(olume)605 b(2)p Fs(,)3336 62147 y(Elsevier)402 b(Science,)h(1995.)p 0 1 0 0 TeXcolorcmyk 942 w(14)p (#page.14) [[218 158 229 170] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 64335 a([20])p Black 650 w(R.)383 b(L.)g(Graham,)g(D.)g (E.)g(Kn)-34 b(uth)385 b(and)f(O.)f(P)-34 b(atashnik,)384 b Fi(Concr)-62 b(ete)414 b(Mathematics)p Fs(,)379 b(Addison-W)-101 b(esley)g(,)383 b(1989.)p 0 1 0 0 TeXcolorcmyk 3336 65530 a(1)p (#page.1) [[102 127 107 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(6)p (#page.6) [[114 127 120 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(7)p (#page.7) [[126 127 132 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 403 w(9)p (#page.9) [[138 127 144 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 67718 a([21])p Black 650 w(F.)404 b(Harary)-101 b(,)403 b Fi(Gr)-62 b(aph)432 b(The)-62 b(ory)p Fs(,)403 b(Addison-W)-101 b(esley)g(,)403 b(1969.)p 0 1 0 0 TeXcolorcmyk 942 w(1)p (#page.1) [[343 108 348 120] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 403 w(14)p (#page.14) [[355 108 366 120] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 69973 a([22])p Black 650 w(T.)330 b(Jenkyns)g(and)h(E.) e(Muller,)g(T)-101 b(riangular)331 b(triples)e(from)h(ceilings)g(to)g (\260o)34 b(ors,)330 b Fi(A)-31 b(mer.)365 b(Math.)f(Monthly)435 b Fm(107)3336 71296 y Fs(\(2000\),)405 b(635{639.)p 0 1 0 0 TeXcolorcmyk 941 w(4)p (#page.4) [[189 75 194 87] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(14)p (#page.14) [[201 75 212 87] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 26150 74617 a FL(16)p Black eop %%Page: 17 17 17 16 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fs([23])p Black 650 w(M.)373 b(J.)f(Marsden,)h(triangles)g(with) h(in)-34 b(teger-v)-67 b(alued)372 b(sides,)h Fi(A)-31 b(mer.)405 b(Math.)e(Monthly)479 b Fm(81)372 b Fs(\(1974\),)i(373{376.) p 0 1 0 0 TeXcolorcmyk 3336 2619 a(4)p (#page.4) [[102 693 107 705] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 4635 a([24])p Black 650 w(R.)404 b(E.)g(Mic)-34 b(k)g(ens,)403 b Fi(Di\256er)-62 b(enc)g(e)433 b(Equations)p Fs(,)402 b(2nd)j(edition,)f(V)-101 b(an)405 b(Nostrand,)g(1990.)p 0 1 0 0 TeXcolorcmyk 941 w(10)p (#page.10) [[448 675 459 687] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(11)p (#page.11) [[466 675 477 687] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 6954 a([25])p Black 650 w(T.)425 b(S.)h(Motzkin)f(and)h (E.)f(G.)g(Straus,)h(Maxima)f(for)h(graphs)g(and)g(a)f(new)h(pro)34 b(of)425 b(of)h(a)f(theorem)h(of)f(T)-101 b(ur\266)-606 b(an,)3336 8277 y Fi(Canad.)433 b(J.)g(Math.)403 b Fm(17)h Fs(\(1965\),)g(533{540.)p 0 1 0 0 TeXcolorcmyk 942 w(14)p (#page.14) [[287 643 298 655] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 10596 a([26])p Black 650 w(E.)500 b(A.)h(Nordhaus)h (and)f(J.)g(W.)f(Gaddum,)i(On)f(complemen)-34 b(tary)501 b(graphs,)g Fi(A)-31 b(mer.)522 b(Math.)f(Monthly)p Fs(,)499 b Fm(63)3336 11919 y Fs(\(1956\),)405 b(175-177.)p 0 1 0 0 TeXcolorcmyk 941 w(6)p (#page.6) [[187 610 192 622] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(13)p (#page.13) [[199 610 210 622] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 14238 a([27])p Black 650 w(P)-101 b(.)562 b(P)-34 b(eart)562 b(and)h(W.-J.)e(W)-101 b(oan,)562 b(Generating)h(functions)g(via)f(Hank)-34 b(el)562 b(and)h(Stieltjes)f (matrices,)g(Article)3336 15493 y(00.2.1,)403 b(V)-101 b(ol.)404 b Fm(3)g Fi(J.)434 b(Inte)-62 b(ger)433 b(Se)-62 b(q.)p Fs(,)403 b(2000,)h(published)h(electronically)d(at)p 0 1 0 0 TeXcolorcmyk 3336 16749 a Fi(http://www.r)-62 b(ese)g(ar)g(ch.att.c)g(om/~njas/se)g(quenc)g(es/JIS/V)-31 b(OL3/p)-62 b(e)g(art1.html)p [[102 566 432 578] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~{}njas/sequences/JIS/VOL3/peart1.html) pdfm Black Fs(.)p 0 1 0 0 TeXcolorcmyk 926 w(6)p (#page.6) [[444 566 449 578] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 19068 a([28])p Black 650 w(G.)298 b(P\266)-606 b(oly)-34 b(a,)298 b(On)g(picture-writing,)g Fi(A)-31 b(mer.)336 b(Math.)f(Monthly)p Fs(,)296 b Fm(63)i Fs(\(1956\),)h (689{697.)g(Reprin)-34 b(ted)299 b(in)f(I.)f(Gessel)3336 20323 y(and)405 b(G.-C.)f(Rota,)g(eds.,)g Fi(Classic)432 b(Pap)-62 b(ers)432 b(in)i(Combinatorics)p Fs(,)401 b(Birkh\304)-606 b(auser,)403 b(1987,)h(249{257.)p 0 1 0 0 TeXcolorcmyk 942 w(15)p (#page.15) [[507 534 518 546] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 22575 a([29])p Black 650 w(J.)k(Riordan,)g Fi(A)-31 b(n)438 b(Intr)-62 b(o)g(duction)435 b(to)i(Combinatorial)e(A) -31 b(nalysis)p Fs(,)406 b(Princeton)i(Univ)-34 b(ersit)g(y)408 b(Press,)f(1978.)p 0 1 0 0 TeXcolorcmyk 957 w(4)p (#page.4) [[539 514 544 526] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 3336 23770 a(14)p (#page.14) [[102 503 113 515] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(15)p (#page.15) [[120 503 131 515] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 26022 a([30])p Black 650 w(C.)508 b(Sc)-34 b(hensted,)510 b(Longest)f(increasing)e(and)j(decreasing)e (subsequences,)g Fi(Canad.)529 b(J.)h(Math.)506 b Fm(13)i Fs(\(1961\),)3336 27277 y(179{191.)670 b(Reprin)-34 b(ted)672 b(in)e(I.)g(Gessel)g(and)h(G.-C.)f(Rota,)i(eds.,)e Fi(Classic)677 b(Pap)-62 b(ers)677 b(in)i(Combinatorics)p Fs(,)3336 28465 y(Birkh\304)-606 b(auser,)403 b(1987,)h(299{311.)p 0 1 0 0 TeXcolorcmyk 941 w(4)p (#page.4) [[239 461 244 473] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 30717 a([31])p Black 650 w(N.)g(J.)f(A.)h(Sloane,)p 0 1 0 0 TeXcolorcmyk 404 w Fi(The)433 b(On-Line)i(Encyclop)-62 b(e)g(dia)430 b(of)j(Inte)-62 b(ger)433 b(Se)-62 b(quenc)g(es)p [[182 441 409 453] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences) pdfm Black Fs(,)402 b(published)j(electronically)d(at)3336 32040 y(h)-34 b(ttp://www.researc)g(h.att.com/~njas/sequences/,)407 b(2000.)p 0 1 0 0 TeXcolorcmyk 941 w(2)p (#page.2) [[368 429 373 441] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(3)p (#page.3) [[380 429 385 441] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(5)p (#page.5) [[392 429 397 441] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(6)p (#page.6) [[404 429 410 441] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 34359 a([32])p Black 650 w(N.)d(J.)f(A.)h(Sloane,)p 0 1 0 0 TeXcolorcmyk 404 w Fi(Classic)432 b(Se)-62 b(quenc)g(es)p [[182 408 266 420] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences/classic.html) pdfm Black Fs(,)402 b(published)j(electronically)d(at)3336 35682 y(h)-34 b(ttp://www.researc)g (h.att.com/~njas/sequences/classic.h)g(tml,)407 b(2000.)p 0 1 0 0 TeXcolorcmyk 941 w(5)p (#page.5) [[423 396 428 408] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 38001 a([33])p Black 650 w(N.)386 b(J.)f(A.)h(Sloane)h (and)g(S.)f(Plou\256e,)f Fi(The)417 b(Encyclop)-62 b(e)g(dia)414 b(of)i(Inte)-62 b(ger)416 b(Se)-62 b(quenc)g(es)p Fs(,)384 b(Academic)i(Press,)f(1995.)p 0 1 0 0 TeXcolorcmyk 3336 39196 a(15)p (#page.15) [[102 364 113 376] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 41212 a([34])p Black 650 w(S.)437 b(E.)f(Sp)34 b(eed,)436 b(The)h(in)-34 b(teger)437 b(sequence)f(A027434)h(and)g(lo) -34 b(w)g(er)437 b(an)-34 b(tagonistic)438 b(functions,)g(in)f (preparation.)p 0 1 0 0 TeXcolorcmyk 3336 42407 a(6)p (#page.6) [[102 335 107 347] [1 1 1 [3 3]] [0 0 1]] pdfm Black Black 800 44423 a([35])p Black 650 w(D.)404 b(Stan)-34 b(ton)406 b(and)f(D.)f(White,)f Fi(Constructive)431 b(Combinatorics)p Fs(,)402 b(Springer-V)-101 b(erlag,)403 b(1986.)p 0 1 0 0 TeXcolorcmyk 941 w(4)p (#page.4) [[484 317 489 329] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 404 w(11)p (#page.11) [[496 317 507 329] [1 1 1 [3 3]] [0 0 1]] pdfm Black 800 47427 52000 45 v 800 49597 a(2000)h Fi(Mathematics)431 b(Subje)-62 b(ct)431 b(Classi\257c)-62 b(ation)p Fs(:)402 b(05A15,)i(05A18,)g(05C35,)g(05C69,)h(05E10,)f(05D05,)f(06B99.)800 50785 y Fi(Keywor)-62 b(ds:)424 b(A)-31 b(ntagonistic)423 b(functions,)g(gr)-62 b(aph)424 b(the)-62 b(ory,)423 b(domination)h(numb)-62 b(er,)424 b(MAPLE,)i(Nor)-62 b(dhaus-Gaddum)800 51973 y(ine)g(quality,)495 b(T)-93 b(ur\266)-620 b(an)-124 b('s)498 b(numb)-62 b(er,)498 b(p)-62 b(artitions)496 b(of)i(inte)-62 b(gers,)497 b(Y)-93 b(oung)497 b(table)-62 b(aux,)496 b(R)-62 b(obinson-Schenste)g(d-Knuth) 494 b(al-)800 53161 y(gorithm)p 800 54770 V 800 57007 a Fs(\(Concerned)405 b(with)g(sequence)p 0 1 0 0 TeXcolorcmyk 404 w(A002620)p 15053 57201 4546 49 v [[207 204 248 216] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A002620) pdfm Black(.\))800 59063 y(Receiv)-34 b(ed)359 b(Jan)-34 b(uary)360 b(10,)g(2001;)g(revised)f(v)-34 b(ersions)359 b(receiv)-34 b(ed)359 b(Marc)-34 b(h)360 b(19,)f(2002;)h(F)-101 b(ebruary)360 b(26,)f(2003.)h(Published)800 60251 y(in)404 b Fi(Journal)433 b(of)g(Inte)-62 b(ger)433 b(Se)-62 b(quenc)g(es)402 b Fs(Marc)-34 b(h)405 b(2,)e(2003.)p Black 26150 74617 a FL(17)p Black eop %%Trailer end end