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b(the)e(ordinary)i(generating)f(functions)g(of)h(the)e (sequences\);)497 b(on)476 b(the)g(other)g(hand,)486 b(the)475 b(L-)-2350 43199 y(sequence)351 b(of)h(the)f(direct)g(pro)36 b(duct)350 b(is)h(obtained)g(as)g(EXPCONV)g(\(here)f(one)i(considers)e (the)h(exp)36 b(onen)-36 b(tial)-2350 44804 y(generating)434 b(functions\).)-399 46409 y(F)-108 b(or)587 b(the)g(P)-36 b(ark)g(er)588 b(sequences)f(the)g(corresp)36 b(onding)587 b(op)36 b(eration)587 b(is)h(simply)g(the)f(sum)g(\(elemen)-36 b(t)587 b(b)-36 b(y)-2350 48015 y(elemen)g(t\):)15426 49620 y Fv(p)16079 49819 y Fs(k)16648 49620 y Fx(\()p Fv(G)295 b Fr(\243)g Fv(H)103 b Fx(\))368 b(=)h Fv(p)23892 49819 y Fs(k)24460 49620 y Fx(\()p Fv(G)p Fx(\))295 b(+)g Fv(p)28753 49819 y Fs(k)29322 49620 y Fx(\()p Fv(H)103 b Fx(\))p Fv(:)-399 51699 y Fx(F)-108 b(orming)587 b(the)g(direct)f (pro)36 b(duct)586 b(of)i Fv(G)f Fx(with)g(the)f(coun)-36 b(table)587 b(symmetric)g(group)g Fv(S)664 b Fx(giv)-36 b(es,)627 b(as)587 b(U-)-2350 53304 y(sequence,)449 b(PSUM)c(of)h(the)f (L-sequence)g(of)h Fv(G)p Fx(;)452 b(as)446 b(L-sequence,)i(BINOMIAL)d (of)i(its)e(L-sequence.)614 b(F)-108 b(or)-2350 54910 y(the)433 b(P)-36 b(ark)g(er)434 b(sequence,)g(it)g(simply)g(yields) 16959 57272 y Fv(p)17612 57471 y Fs(k)18181 57272 y Fx(\()p Fv(G)295 b Fr(\243)g Fv(S)77 b Fx(\))369 b(=)g Fv(p)25122 57471 y Fs(k)25690 57272 y Fx(\()p Fv(G)p Fx(\))295 b(+)g(1)p Fv(:)-399 59634 y Fx(One)384 b(ma)-36 b(y)385 b(also)g(consider)f(the)f (pro)36 b(duct)383 b(action)i(of)g Fv(G)194 b Fr(\243)g Fv(H)487 b Fx(on)384 b(the)f(cartesian)i(pro)36 b(duct)383 b Fv(X)299 b Fr(\243)194 b Fv(Y)289 b Fx(.)562 b(F)-108 b(or)-2350 61239 y(this)434 b(action)g(one)f(has:)14027 62844 y Fv(p)14680 63043 y Fs(k)15249 62844 y Fx(\()p Fv(G)294 b Fr(\243)i Fv(H)103 b Fx(\))368 b(=)23107 61582 y Fq(X)23560 64492 y Fs(i;j)21840 65427 y Ft(lcm\()p Fs(i;j)51 b Ft(\)=)p Fs(k)26515 62844 y Fv(p)27168 63043 y Fs(i)27544 62844 y Fx(\()p Fv(G)p Fx(\))p Fv(p)30235 63043 y Fs(j)30721 62844 y Fx(\()p Fv(H)103 b Fx(\))p Fv(:)-399 67363 y Fx(What)444 b(happ)36 b(ens)443 b(for)h(wreath)g(pro) 36 b(ducts)442 b(is)i(more)g(in)-36 b(teresting.)609 b(Recall)445 b([)p 0 1 0 0 TeXcolorcmyk(7)p (#cite.Gew02) [[395 111 401 123] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 444 w(8)p (#cite.GeMe01) [[408 111 414 123] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])f(that)f(for)h(the)f(P)-36 b(ark)g(er)-2350 68968 y(sequences)434 b(of)g(the)f(wreath)h(pro)36 b(duct)432 b(of)j Fv(G)e Fx(and)g Fv(H)536 b Fx(the)433 b(follo)-36 b(wing)436 b(holds:)15104 71489 y Fv(p)15757 71688 y Fs(k)16325 71489 y Fx(\()p Fv(G)295 b Fr(o)g Fv(H)103 b Fx(\))368 b(=)22252 70227 y Fq(X)22582 73105 y Fs(d)p Fp(j)p Fs(k)24392 71489 y Fv(p)25045 71688 y Fs(d)25585 71489 y Fx(\()p Fv(G)p Fx(\))p Fv(p)28276 71695 y Fs(k)24 b(=d)29798 71489 y Fx(\()p Fv(G)p Fx(\))p Fv(:)p Black Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black 41095 -272 a Fx(3)p Black -2350 3824 a(This)367 b(is)f(the)g(Diric)-36 b(hlet)366 b(con)-36 b(v)g(olution,)381 b(whic)-36 b(h)366 b(in)g(the)f (terminology)j(of)e(Sloane)h([)p 0 1 0 0 TeXcolorcmyk(13)p (#cite.transf) [[403 683 415 695] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])f(is)g(the)g(DIRICHLET)-2350 5429 y(transform)434 b(of)g(the)f(t)-36 b(w)g(o)434 b(sequences.)-399 7034 y(W)-108 b(e)418 b(ma)-36 b(y)418 b(no)-36 b(w)417 b(study)-108 b(,)420 b(for)e(a)g(giv)-36 b(en)418 b(oligomorphic)h(group)e Fv(H)103 b Fx(,)420 b(the)d(op)36 b(erator)417 b(mapping)g(the)g(P)-36 b(ark)g(er)-2350 8639 y(sequence)433 b(of)h(an)-36 b(y)433 b(group)f Fv(G)h Fx(to)g(that)f(of)h Fv(G)294 b Fr(o)f Fv(H)103 b Fx(.)578 b(F)-108 b(or)432 b(U-sequences,)h(this)f(pro)36 b(cedure)432 b(giv)-36 b(es)434 b(rise)f(to)g(the)-2350 10244 y(op)36 b(erators)502 b(EULER,)g(INVER)-108 b(T,)502 b(and)f(CIK,)h(resp)36 b(ectiv)-36 b(ely)503 b(for)f Fv(H)586 b Fx(=)485 b Fv(S)77 b Fx(,)519 b Fv(H)586 b Fx(=)484 b Fv(A)p Fx(,)519 b(and)501 b Fv(H)587 b Fx(=)484 b Fv(C)95 b Fx(.)782 b(F)-108 b(or)-2350 11850 y(P)-36 b(ark)g(er)434 b(sequences)g(w)-36 b(e)434 b(get,)g(for)g Fv(H)471 b Fx(=)369 b Fv(S)77 b Fx(,)434 b(the)f(MOBIUSi)g(op)36 b(erator)17362 14816 y Fv(p)18015 15015 y Fs(k)18584 14816 y Fx(\()p Fv(G)295 b Fr(o)g Fv(S)77 b Fx(\))369 b(=)24208 13554 y Fq(X)24537 16432 y Fs(d)p Fp(j)p Fs(k)26348 14816 y Fv(p)27001 15015 y Fs(d)27540 14816 y Fx(\()p Fv(G)p Fx(\);)-2350 19022 y(and,)434 b(for)g Fv(H)471 b Fx(=)369 b Fv(A)p Fx(,)434 b(the)f(iden)-36 b(tit)g(y)433 b(op)36 b(erator)18369 21830 y Fv(p)19022 22029 y Fs(k)19590 21830 y Fx(\()p Fv(G)295 b Fr(o)g Fv(A)p Fx(\))369 b(=)g Fv(p)25965 22029 y Fs(k)26533 21830 y Fx(\()p Fv(G)p Fx(\))p Fv(:)-2350 24637 y Fx(F)-108 b(or)434 b Fv(H)471 b Fx(=)368 b Fv(C)529 b Fx(w)-36 b(e)434 b(get)15332 26401 y Fv(p)15985 26600 y Fs(k)16554 26401 y Fx(\()p Fv(G)295 b Fr(o)g Fv(C)95 b Fx(\))369 b(=)22326 25139 y Fq(X)22656 28017 y Fs(d)p Fp(j)p Fs(k)24467 26401 y Fv(p)25120 26600 y Fs(d)25659 26401 y Fx(\()p Fv(G)p Fx(\))p Fv(')p Fx(\()p Fv(k)45 b(=d)p Fx(\);)-2350 30144 y(in)479 b(particular)f(note)g(that)g(for)g(square-free)h Fv(k)45 b Fx('s)479 b(\(that)e(is,)490 b(the)478 b(v)-72 b(alues)479 b(of)g Fv(k)523 b Fx(suc)-36 b(h)477 b(that)h Fv(\271)p Fx(\()p Fv(k)45 b Fx(\))444 b Fr(6)p Fx(=)h(0\))478 b(one)-2350 31749 y(has)434 b Fv(p)623 31948 y Fs(k)1192 31749 y Fx(\()p Fv(G)295 b Fr(o)g Fv(C)95 b Fx(\))368 b(=)h Fv(')p Fx(\()p Fv(k)45 b Fx(\))9770 30753 y Fq(P)11172 32136 y Fs(d)p Fp(j)p Fs(k)12708 31749 y Fv(p)13361 31948 y Fs(d)13900 31749 y Fx(\()p Fv(G)p Fx(\))p Fv(=')p Fx(\()p Fv(d)p Fx(\).)-399 33354 y(Notice)531 b(that,)553 b(while)531 b(in)e(general)i Fv(G)360 b Fr(o)h Fv(H)632 b Fx(and)529 b Fv(H)463 b Fr(o)361 b Fv(G)529 b Fx(ma)-36 b(y)531 b(b)36 b(e)529 b(di\256eren)-36 b(t)529 b(groups,)553 b(they)530 b(ha)-36 b(v)g(e)530 b(the)-2350 34959 y(same)434 b(P)-36 b(ark)g(er)434 b(sequence;)g(so)g(these)f(op)36 b(erators)434 b(are)g(also)g(those)g(mapping)f Fu(p)p Fx(\()p Fv(G)p Fx(\))g(to)h Fu(p)p Fx(\()p Fv(H)397 b Fr(o)295 b Fv(G)p Fx(\).)-2350 39378 y Fy(3)2152 b(P)-60 b(ark)g(er)685 b(sequences)e(and)g(circulan)-60 b(t)684 b(relational)f(structures)-2350 42298 y Fx(Recall)450 b([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.GeMe01) [[90 336 95 348] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])e(that,)k(if)d(w)-36 b(e)448 b(are)g(dealing)h(with)f(a)h (group)f Fv(G)g Fx(de\257ned)e(as)j(the)f(automorphism)f(group)h(of)h (the)-2350 43903 y(limit)360 b(of)f(a)g(F)-108 b(ra)-145 b(\304)-506 b(\263ss)-36 b(\266)-614 b(e)359 b(class)h Fr(F)490 b Fx(of)359 b(relational)h(structures,)373 b(the)358 b(P)-36 b(ark)g(er)358 b(sequence)h(of)g Fv(G)g Fx(has)f(an)h (alternativ)-36 b(e)-2350 45508 y(in)g(terpretation)360 b(as)h(the)f(sequence)g(en)-36 b(umerating)360 b(the)g(\257nite)g (circulan)-36 b(t)360 b(structures)f(in)i(that)f(class.)554 b(More)-2350 47114 y(precisely)-108 b(,)699 b(the)645 b Fv(k)45 b Fx(th)645 b(comp)36 b(onen)-36 b(t)645 b(of)h(the)f(P)-36 b(ark)g(er)645 b(sequence)g(coun)-36 b(ts)645 b(the)g(relational)i (structures)d(in)-2350 48719 y(\(the)f(age)h(of)93 b(\))644 b Fr(F)775 b Fx(on)644 b(the)f(set)g Fr(f)p Fx(1)p Fv(;)221 b Fx(2)p Fv(;)g(:)g(:)g(:)k(;)c(k)45 b Fr(g)645 b Fx(admitting)e(as)h (an)f(automorphism)g(the)g(p)36 b(erm)-36 b(utation)-2350 50324 y(\(1)600 b(2)822 b Fv(:)221 b(:)g(:)823 b(k)45 b Fx(\))599 b(\(note)h(that)f(this)g(is)h(di\256eren)-36 b(t)599 b(than)g(just)h(requesting)g(that)f(the)g(structure)f(admits)i (a)-2350 51929 y(circular)450 b(symmetry\).)628 b(In)449 b(what)h(follo)-36 b(ws)452 b(w)-36 b(e)450 b(shall)g(use)g(\\circulan) -36 b(t)450 b([structure]")f(to)h(mean)f(\\circulan)-36 b(t)-2350 53534 y([structure])570 b(on)h(the)f(set)h Fr(f)p Fx(1)p Fv(;)221 b Fx(2)p Fv(;)g(:)g(:)g(:)k(;)c(k)45 b Fr(g)571 b Fx(admitting)g(the)f(automorphism)g(\(1)h(2)793 b Fv(:)221 b(:)g(:)793 b(k)45 b Fx(\)".)990 b(All)572 b(of)f(the)-2350 55139 y(P)-36 b(ark)g(er)361 b(sequences)f(listed)h (in)f(the)g(\\F)-108 b(ra)-145 b(\304)-506 b(\263ss)-36 b(\266)-614 b(e)362 b(class")f(table)g(w)-36 b(ere)360 b(obtained)g(b)-36 b(y)361 b(coun)-36 b(ting)360 b(these)g(circulan)-36 b(t)-2350 56744 y(structures.)-399 58349 y(This)389 b(mirrors)e(what)h (happ)36 b(ens)387 b(with)h(the)f(L-sequence)g(\()p Fv(F)28351 58548 y Fs(k)28920 58349 y Fx(\))g(of)i(the)e(same)h(group,)397 b(whic)-36 b(h)388 b(is)g(de\257ned)-2350 59954 y(as)601 b(the)f(n)-36 b(um)g(b)36 b(er)599 b(of)j(orbits)e(on)h Fv(k)45 b Fx(-tuples)599 b(of)i(distinct)f(elemen)-36 b(ts,)643 b(and)600 b(is)h(equal)g(to)g(the)f(n)-36 b(um)g(b)36 b(er)599 b(of)-2350 61559 y(lab)36 b(elled)566 b(structures)d(on)h Fv(k)610 b Fx(p)36 b(oin)-36 b(ts.)971 b(The)564 b(same)h(holds)g(for)g (the)f(U-sequence)g Fv(f)38622 61758 y Fs(k)39755 61559 y Fx(of)i(the)e(n)-36 b(um)g(b)36 b(er)563 b(of)-2350 63164 y(orbits)431 b(on)g Fv(k)45 b Fx(-sets,)432 b(giving)g(the)f(n) -36 b(um)g(b)36 b(er)429 b(of)j(unlab)36 b(elled)431 b(structures.)576 b(The)432 b(theory)f(b)36 b(ehind)430 b(this)h(can)g(b)36 b(e)-2350 64770 y(found)434 b(in)f(Cameron's)h(b)36 b(o)g(ok)435 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.Cam90) [[187 134 193 146] [1 1 1 [3 3]] [0 0 1]] pdfm Black(].)-399 66375 y(In)495 b(order)g(to)g(giv)-36 b(e)497 b(an)e(idea)g(of)h(the)e(tec)-36 b(hniques)495 b(in)-36 b(v)g(olv)g(ed)496 b(in)f(deriving)g(P)-36 b(ark)g(er)496 b(sequences,)511 b(let)495 b(us)-2350 67980 y(\257rst)433 b(brie\260y)h(recall)g([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.GeMe01) [[146 105 151 117] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(what)f(happ)36 b(ens)433 b(for)h(graphs.)-399 69585 y(T)-108 b(o)362 b(describ)36 b(e)362 b(a)g(circulan)-36 b(t)361 b(graph)g(\241)h(on)f(the)g(v)-36 b(ertex)363 b(set)e Fr(f)p Fx(0)p Fv(;)221 b Fx(1)p Fv(;)g Fx(2)p Fv(;)g(:)g(:)g(:)227 b(;)221 b(k)193 b Fr(\241)148 b Fx(1)p Fr(g)p Fx(,)378 b(it)362 b(is)g(su\261cien)-36 b(t)361 b(to)g(giv)-36 b(e)-2350 71190 y(the)405 b(neigh)-36 b(b)36 b(ours)405 b(of)h(a)f(\257xed)g(v)-36 b(ertex)406 b(\(sa)-36 b(y)406 b(0\);)415 b(this)405 b(subset,)410 b(whic)-36 b(h)405 b(has)g(the)g(prop)36 b(ert)-36 b(y)404 b(that)h(it)h(con)-36 b(tains)-2350 72795 y(a)380 b(v)-36 b(ertex)379 b Fv(i)g Fx(if)g(and)f(only)i(if)g(it)f(con)-36 b(tains)378 b Fv(k)229 b Fr(\241)183 b Fv(i)p Fx(,)390 b(is)379 b(called)h Fw(symb)-66 b(ol)516 b Fx(of)380 b(\241.)559 b(On)378 b(the)h(other)f(hand)g(an)-36 b(y)379 b(subset)p Black Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black 41095 -272 a Fx(4)p Black -2350 3824 a Fv(S)512 b Fx(of)435 b Fr(f)p Fx(1)p Fv(;)221 b Fx(2)p Fv(;)g(:)g(:)g(:)k(;)c(k)341 b Fr(\241)296 b Fx(1)p Fr(g)435 b Fx(suc)-36 b(h)433 b(that)h Fv(i)370 b Fr(2)f Fv(S)512 b Fx(implies)435 b Fv(k)340 b Fr(\241)296 b Fv(i)370 b Fr(2)f Fv(S)512 b Fx(is)434 b(a)h(p)36 b(ossible)435 b(sym)-36 b(b)36 b(ol)434 b(for)h(a)g(graph.)580 b(So)-2350 5429 y(the)450 b Fv(k)45 b Fx(th)450 b(en)-36 b(try)451 b(of)g(the)f(P)-36 b(ark)g(er)451 b(sequence)f(of)h(the)f(automorphism) g(group)h(of)g(the)f(limit)h(of)g(the)f(F)-108 b(ra)-145 b(\304)-506 b(\263ss)-36 b(\266)-614 b(e)-2350 7034 y(class)435 b(of)f(graphs)f(\(that)g(is)h(the)f(w)-36 b(ell-kno)g(wn)434 b(random,)g(or)g(Erd})-650 b(os-R)-36 b(\266)-614 b(en)-36 b(yi,)433 b(or)h(Rado,)g(graph\))f(is)h(2)46016 6552 y Fp(b)p Fs(k)24 b(=)p Ft(2)p Fp(c)48363 7034 y Fx(.)-399 8639 y(Sev)-36 b(eral)506 b(v)-72 b(ariations)506 b(to)g(this)e(metho) 36 b(d)505 b(yield)h(the)e(P)-36 b(ark)g(er)506 b(sequences)f(for)h (other)e(relational)j(struc-)-2350 10244 y(tures.)-399 11850 y(F)-108 b(or)481 b(instance,)492 b(if)482 b(w)-36 b(e)481 b(consider)f(the)g(sym)-36 b(b)36 b(ol)482 b(for)f(a)g(digraph) f(\(a)h(structure)e(with)i(a)g(relation)g Fr(!)h Fx(in)-2350 13455 y(whic)-36 b(h)471 b(for)h(eac)-36 b(h)470 b(pair)h(of)h (distinct)e(v)-36 b(ertices)472 b Fv(a)p Fx(,)480 b Fv(b)p Fx(,)g(an)-36 b(y)471 b(of)h Fv(a)432 b Fr(!)h Fv(b)p Fx(,)481 b Fv(b)432 b Fr(!)h Fv(a)p Fx(,)480 b(b)36 b(oth,)480 b(or)471 b(none)f(ma)-36 b(y)472 b(hold\))-2350 15060 y(w)-36 b(e)486 b(c)-36 b(ho)36 b(ose)486 b(whether)f(or)h(not)f(to)h (join,)499 b(b)-36 b(y)486 b(putting)e(a)i(directed)f(edge,)499 b(0)486 b(with)g(an)-36 b(y)485 b(other)h(v)-36 b(ertex.)735 b(So)-2350 16665 y(w)-36 b(e)477 b(get)f Fv(p)2471 16864 y Fs(k)3482 16665 y Fx(=)441 b(4)5585 16183 y Ft(\()p Fs(k)24 b Fp(\241)p Ft(1\))p Fs(=)p Ft(2)9471 16665 y Fx(=)441 b(2)11574 16183 y Fs(k)24 b Fp(\241)p Ft(1)13346 16665 y Fx(.)706 b(Similarly)-108 b(,)488 b(if)477 b(w)-36 b(e)477 b(do)f(not)g(allo)-36 b(w)478 b(a)f(double)e(orien)-36 b(tation)477 b(on)f(an)g(edge,)-2350 18270 y(w)-36 b(e)434 b(get)g(the)f(class)h(of)h(orien)-36 b(ted)433 b(graphs,)g(for)h(whic) -36 b(h)434 b Fv(p)24447 18469 y Fs(k)25384 18270 y Fx(=)369 b(3)27415 17788 y Fp(b)p Fs(k)24 b(=)p Ft(2)p Fp(c)29762 18270 y Fx(.)-399 19875 y(Of)522 b(course,)543 b(this)521 b(kind)g(of)h(argumen)-36 b(t)521 b(holds)g(also)h(for)g(the)e(class)i (of)g Fv(n)p Fx(-ary)g(relations,)544 b(for)522 b Fv(n)c Fr(\270)h Fx(2.)-2350 21480 y(The)474 b(sym)-36 b(b)36 b(ol)475 b(for)f(a)g(circulan)-36 b(t)473 b Fv(n)p Fx(-relation)i(on)e Fv(k)519 b Fx(p)36 b(oin)-36 b(ts)473 b(can)h(b)36 b(e)474 b(an)-36 b(y)474 b(p)36 b(ossible)474 b(set)f(of)i(\()p Fv(n)323 b Fr(\241)f Fx(1\)-tuples)-2350 23085 y(\(admitting)372 b(rep)36 b(etitions\))372 b(of)h(the)f(p)36 b(oin)-36 b(ts.)558 b(F)-108 b(or)371 b(instance,)385 b(for)373 b(a)f(ternary)g(relation,)386 b(w)-36 b(e)372 b(ma)-36 b(y)373 b(ha)-36 b(v)g(e)372 b(\(0)p Fv(;)221 b Fx(0\))-2350 24690 y(\(meaning)467 b(that)f(\(0)p Fv(;)221 b Fx(0)p Fv(;)g Fx(0\))469 b(holds\),)474 b(\(0)p Fv(;)221 b Fx(1\),)476 b(\(1)p Fv(;)221 b Fx(0\),)477 b(\(1)p Fv(;)221 b Fx(1\),)476 b(.)217 b(.)g(.)683 b(So)467 b(w)-36 b(e)467 b(ha)-36 b(v)g(e)466 b Fv(k)36083 24208 y Fs(n)p Fp(\241)p Ft(1)38379 24690 y Fx(suc)-36 b(h)465 b(\()p Fv(n)318 b Fr(\241)g Fx(1\)-tuples,)-2350 26295 y(and)434 b(2)830 25813 y Fs(k)1344 25501 y Fo(n)p Fn(\241)p Fm(1)3444 26295 y Fx(p)36 b(ossible)434 b(sym)-36 b(b)36 b(ols)434 b(\(sets)g(of)g(suc) -36 b(h)433 b(tuples\).)-399 27901 y(More)h(examples)g(in)g(same)g(v) -36 b(ein)434 b(app)36 b(ear)433 b(in)g(the)g(tables.)-399 29506 y(The)300 b(same)f(tec)-36 b(hniques)299 b(can)g(b)36 b(e)299 b(applied)g(to)g(the)g(class)h(of)g(t)-36 b(w)g(o-graphs;)344 b(this)299 b(case,)327 b(ho)-36 b(w)g(ev)g(er,)326 b(requires)-2350 31111 y(some)434 b(care.)-399 32716 y(Recall)461 b(that)e(a)h Fw(two-gr)-66 b(aph)559 b Fx(is)460 b(de\257ned)e(as)i(a)g(pair)g(\()p Fv(X)32 b(;)221 b(T)181 b Fx(\),)467 b(where)459 b Fv(X)565 b Fx(is)460 b(a)g(set)f(of)h(p)36 b(oin)-36 b(ts,)466 b(and)460 b Fv(T)640 b Fx(a)-2350 34321 y(set)445 b(of)h(3-subsets)e (of)h Fv(X)550 b Fx(with)445 b(the)g(prop)36 b(ert)-36 b(y)444 b(that)g(an)-36 b(y)445 b(4-subset)f(of)i Fv(X)550 b Fx(con)-36 b(tains)445 b(an)f(ev)-36 b(en)445 b(n)-36 b(um)g(b)36 b(er)444 b(of)-2350 35926 y(mem)-36 b(b)36 b(ers)433 b(of)i Fv(T)181 b Fx(.)-399 37531 y(Tw)-36 b(o-graphs)444 b(on)f Fv(k)488 b Fx(v)-36 b(ertices)444 b(are)f(in)h(bijection)f(with)h(switc)-36 b(hing)443 b(classes)h(of)g(graphs)f(on)g Fv(k)489 b Fx(v)-36 b(ertices.)-2350 39136 y(Recall)458 b(that)f(switc)-36 b(hing)457 b(a)g(graph)f(\241)408 b(=)g(\()p Fv(V)72 b(;)221 b(E)78 b Fx(\))457 b(with)g(resp)36 b(ect)456 b(to)h Fv(S)486 b Fr(\265)408 b Fv(V)746 b Fx(giv)-36 b(es)458 b(a)f(graph)f(\()p Fv(V)72 b(;)221 b(E)45876 38654 y Fp(0)46188 39136 y Fx(\))456 b(suc)-36 b(h)-2350 40741 y(that)380 b Fr(f)p Fv(v)48 b(;)221 b(w)36 b Fr(g)369 b(2)g Fv(E)6633 40259 y Fp(0)7323 40741 y Fx(if)381 b(and)e(only)i(if)f(either)g Fv(v)428 b Fx(and)379 b Fv(w)416 b Fx(are)380 b(b)36 b(oth)379 b(in)h Fv(S)458 b Fx(or)380 b(b)36 b(oth)379 b(in)h Fv(V)475 b Fr(n)186 b Fv(S)457 b Fx(and)380 b Fr(f)p Fv(v)48 b(;)221 b(w)36 b Fr(g)369 b(2)f Fv(E)78 b Fx(,)-2350 42346 y(or)424 b(one)g(is)g(in)g Fv(S)501 b Fx(and)424 b(the)f(other)h(is)g(in)f Fv(V)565 b Fr(n)276 b Fv(S)77 b Fx(,)426 b(and)d Fr(f)p Fv(v)48 b(;)221 b(w)36 b Fr(g)369 b(62)g Fv(E)501 b Fx(\(see)424 b([)p 0 1 0 0 TeXcolorcmyk(11)p (#cite.Sei76) [[366 336 378 348] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)j(also)e(for)f(the)f(description)h(of)-2350 43951 y(the)433 b(corresp)36 b(ondence)433 b(b)36 b(et)-36 b(w)g(een)433 b(t)-36 b(w)g(o-graphs)433 b(and)g(switc)-36 b(hing)434 b(classes\).)-399 45557 y(Note)534 b(that)f(a)g(t)-36 b(w)g(o-graph)533 b(\()p Fv(X)32 b(;)221 b(T)181 b Fx(\))534 b(is)f(circulan)-36 b(t)533 b(if)h(and)f(only)h(if)g(at)f(least)h(one)f (graph)g(in)g(the)g(cor-)-2350 47162 y(resp)36 b(onding)552 b(switc)-36 b(hing)553 b(class)g(is.)936 b(In)552 b(fact,)583 b(assume)553 b(that)f Fv(\256)561 b Fx(is)553 b(a)f(p)36 b(erm)-36 b(utation)552 b(of)h Fv(X)658 b Fx(inducing)552 b(an)-2350 48767 y(automorphism)619 b(of)g(\()p Fv(X)32 b(;)221 b(T)181 b Fx(\);)713 b(then)618 b Fv(\256)627 b Fx(induces)618 b(an)h(automorphism)f(of)i(at)f(least)g(one)g(graph)f (in)h(the)-2350 50372 y(corresp)36 b(onding)407 b(switc)-36 b(hing)407 b(class)h(\(as)f(pro)-36 b(v)g(ed)407 b(b)-36 b(y)407 b(Mallo)-36 b(ws)408 b(and)e(Sloane)i([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.MaSlo75) [[385 264 397 276] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(];)416 b(see)407 b(also)h(Cameron)f([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.cam77) [[502 264 508 276] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\).)-399 51977 y(The)434 b(follo)-36 b(wing)436 b(result)d(relates)h(circulan)-36 b(t)433 b(t)-36 b(w)g(o-graphs)433 b(to)h(circulan)-36 b(t)433 b(graphs.)p Black -2350 55021 a Fu(Theorem)500 b(3.1)p Black 651 w Fw(L)-66 b(et)473 b Fx(\241)g Fw(b)-66 b(e)474 b(a)g(cir)-66 b(culant)472 b Fv(k)45 b Fw(-vertex)473 b(gr)-66 b(aph.)625 b(If)473 b Fv(k)519 b Fw(is)474 b(o)-66 b(dd,)476 b(then)d Fx(\241)h Fw(is)g(the)g(only)g(cir)-66 b(culant)-2350 56626 y(gr)g(aph)364 b(in)e(its)h(switching)f(class;)398 b(if)362 b Fv(k)408 b Fw(is)363 b(even,)383 b(ther)-66 b(e)362 b(ar)-66 b(e)363 b(exactly)f(two)i(cir)-66 b(culant)362 b(gr)-66 b(aphs)363 b(in)g(its)g(switching)-2350 58231 y(class.)-399 61275 y Fx(In)443 b(order)f(to)h(pro)-36 b(v)g(e)443 b(this,)i(let)e(us)f (\257rst)g(sho)-36 b(w)443 b(in)g(some)g(detail)g(what)g(happ)36 b(ens)442 b(switc)-36 b(hing)443 b(circulan)-36 b(t)-2350 62880 y(and)434 b(regular)f(graphs.)p Black -2350 65925 a Fu(Prop)42 b(osition)500 b(3.2)p Black 651 w Fw(F)-100 b(or)427 b Fv(k)471 b Fw(o)-66 b(dd,)433 b(in)425 b(e)-66 b(ach)426 b(switching)f(class)i(of)e(gr)-66 b(aphs)427 b(on)e Fv(k)471 b Fw(vertic)-66 b(es)425 b(ther)-66 b(e)425 b(is)h(at)g(most)-2350 67530 y(one)465 b(r)-66 b(e)g(gular)464 b(gr)-66 b(aph.)-2350 70574 y Fu(Pro)42 b(of.)746 b Fx(Let)489 b(\241)g(b)36 b(e)489 b(a)h(regular)g(graph)f(of)h(v)-72 b(alency)490 b Fv(r)526 b Fx(on)489 b Fv(k)535 b Fx(v)-36 b(ertices.)746 b(Let)489 b(us)g(switc)-36 b(h)490 b(it)f(with)h(resp)36 b(ect)-2350 72179 y(to)589 b(the)e(set)h Fv(S)710 b Fr(\265)632 b Fv(V)289 b Fx(\(\241\),)627 b(0)632 b Fv(<)g Fr(j)p Fv(S)77 b Fr(j)632 b Fx(=)g Fv(m)g(<)f(k)45 b Fx(.)1043 b(Then,)626 b(for)589 b(eac)-36 b(h)588 b Fv(t)632 b Fr(62)f Fv(S)77 b Fx(,)628 b(call)589 b Fv(n)40337 72378 y Fs(t)41321 72179 y Fx(the)e(n)-36 b(um)g(b)36 b(er)587 b(of)p Black Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black 41095 -272 a Fx(5)p Black -2350 3824 a(neigh)-36 b(b)36 b(ours)583 b(of)h Fv(t)f Fx(included)f(in)h Fv(S)660 b Fx(\(b)36 b(efore)584 b(switc)-36 b(hing\).)1027 b(Then)583 b(the)f(v)-72 b(alency)584 b(of)g Fv(t)f Fx(in)g(the)g(switc)-36 b(hed)-2350 5429 y(graph)479 b(is)f Fv(r)362 b Fr(\241)326 b Fv(n)5816 5628 y Fs(t)6538 5429 y Fx(+)f(\()p Fv(m)g Fr(\241)h Fv(n)11979 5628 y Fs(t)12375 5429 y Fx(\).)713 b(Analogously)-108 b(,)491 b(if)479 b Fv(s)445 b Fr(2)g Fv(S)556 b Fx(and)478 b Fv(n)29973 5628 y Fs(s)30942 5429 y Fx(is)h(the)f(n)-36 b(um)g(b)36 b(er)477 b(of)i(neigh)-36 b(b)36 b(ours)478 b(of)h Fv(s)-2350 7034 y Fx(not)434 b(in)f Fv(S)77 b Fx(,)434 b(the)f(v)-72 b(alency)435 b(of)f Fv(s)f Fx(in)h(the)f(switc) -36 b(hed)433 b(graph)g(is)h Fv(r)332 b Fr(\241)295 b Fv(n)29635 7233 y Fs(s)30421 7034 y Fx(+)g(\()p Fv(k)340 b Fr(\241)295 b Fv(m)g Fr(\241)h Fv(n)38116 7233 y Fs(s)38606 7034 y Fx(\).)-399 8639 y(Therefore,)396 b(if)387 b(the)e(switc)-36 b(hed)385 b(graph)g(is)h(regular,)395 b(giv)-36 b(en)386 b(t)-36 b(w)g(o)386 b(v)-36 b(ertices)386 b Fv(s)f Fx(and)g Fv(t)g Fx(as)h(ab)36 b(o)-36 b(v)g(e,)396 b(their)385 b(new)-2350 10244 y(v)-72 b(alencies)435 b(m)-36 b(ust)433 b(b)36 b(e)433 b(equal:)11364 13178 y Fv(r)331 b Fr(\241)295 b Fv(n)14385 13377 y Fs(t)15076 13178 y Fx(+)g(\()p Fv(m)g Fr(\241)g Fv(n)20426 13377 y Fs(t)20822 13178 y Fx(\))368 b(=)h Fv(r)331 b Fr(\241)296 b Fv(n)26099 13377 y Fs(s)26885 13178 y Fx(+)f(\()p Fv(k)340 b Fr(\241)295 b Fv(m)g Fr(\241)g Fv(n)34579 13377 y Fs(s)35070 13178 y Fx(\))p Fv(;)-2350 16111 y Fx(or,)18003 17716 y Fv(k)414 b Fx(=)368 b(2\()p Fv(m)295 b Fr(\241)h Fv(n)25167 17915 y Fs(t)25857 17716 y Fx(+)f Fv(n)27940 17915 y Fs(s)28431 17716 y Fx(\))p Fv(:)-2350 20041 y Fx(That)423 b(is,)i(the)d(n)-36 b(um)g(b)36 b(er)421 b(of)i(v)-36 b(ertices)423 b(m)-36 b(ust)422 b(b)36 b(e)422 b(ev)-36 b(en)423 b(for)g(a)g(non-trivial)g(switc)-36 b(hing)422 b(equiv)-72 b(alence)424 b(to)f(hold)-2350 21646 y(b)36 b(et)-36 b(w)g(een)433 b(\241)h(and)f(another)g(regular)h (graph.)29379 b Fr(})-399 23915 y Fx(W)-108 b(e)612 b(ha)-36 b(v)g(e)613 b(no)-36 b(w)612 b(the)f(\257rst)g(part)h(of)g(the)g (theorem)f(\(b)36 b(ecause)612 b(an)-36 b(y)612 b(circulan)-36 b(t)612 b(graph)g(m)-36 b(ust)611 b(b)36 b(e,)656 b Fw(a)-2350 25521 y(fortiori)p Fx(,)479 b(regular\).)690 b(F)-108 b(or)470 b(the)g(second)g(part,)480 b(the)470 b(follo)-36 b(wing)472 b(prop)36 b(osition)471 b(describ)36 b(es)471 b(explicitely)h(when)-2350 27126 y(switc)-36 b(hing)434 b(a)g(circulan)-36 b(t)433 b(graph)h(yields)g(another)f(circulan)-36 b(t)433 b(graph.)p Black -2350 30170 a Fu(Prop)42 b(osition)500 b(3.3)p Black 651 w Fw(If)594 b Fx(\241)g Fw(is)h(a)f(cir)-66 b(culant)594 b(gr)-66 b(aph)594 b(on)g(the)h(vertic)-66 b(es)592 b Fr(f)p Fx(1)p Fv(;)221 b Fx(2)p Fv(;)g(:)g(:)g(:)226 b(;)221 b(k)45 b Fr(g)p Fw(,)627 b Fv(k)640 b Fw(even,)625 b(the)595 b(only)-2350 31775 y(non-trivial)611 b(switching)g(yielding)g (a)h(cir)-66 b(culant)611 b(gr)-66 b(aph)612 b(is)g(with)g(r)-66 b(esp)g(e)g(ct)611 b(to)h(the)g(set)g(of)g(vertic)-66 b(es)611 b Fv(S)719 b Fx(=)-2350 33380 y Fr(f)p Fx(1)p Fv(;)221 b Fx(3)p Fv(;)g Fx(5)p Fv(;)g(:)g(:)g(:)227 b(;)221 b(k)341 b Fr(\241)295 b Fx(1)p Fr(g)465 b Fw(\(or)h(its)e(c)-66 b(omplement\).)-2350 36424 y Fu(Pro)42 b(of.)968 b Fx(F)-108 b(or)563 b(\241)g(to)h(b)36 b(e)563 b(circulan)-36 b(t,)596 b(it)563 b(m)-36 b(ust)563 b(b)36 b(e)563 b(p)36 b(ossible)564 b(to)g(decomp)36 b(ose)563 b(it)h(in)f(cycles)h(\()p Fv(i;)221 b(i)384 b Fx(+)f Fv(l)29 b(;)221 b(i)385 b Fx(+)-2350 38029 y(2)p Fv(l)29 b(;)221 b(:)g(:)g(:)k(;)c(i)369 b Fr(\241)g Fv(l)29 b Fx(\))542 b(\(all)g(additions)f(mo)36 b(dulo)542 b Fv(k)45 b Fx(\).)901 b(In)541 b(eac)-36 b(h)542 b(suc)-36 b(h)540 b(cycle)i(the)f(v)-36 b(ertices)542 b(either)f(ha)-36 b(v)g(e)542 b(all)g(the)-2350 39634 y(same)433 b(parit)-36 b(y)-108 b(,)432 b(or)g(an)g(o)36 b(dd)432 b(and)g(an)g(ev)-36 b(en)432 b(v)-36 b(ertex)433 b(alternate.)578 b(So,)432 b(switc)-36 b(hing)432 b(with)h(resp)36 b(ect)431 b(to)h Fv(S)510 b Fx(either)-2350 41239 y(preserv)-36 b(es)391 b(the)g(whole)h(cycle,)401 b(or)391 b(causes)g(all)i(its)e (edges)g(to)g(v)-72 b(anish.)565 b(In)391 b(either)g(case,)400 b(the)391 b(graph)f(remains)-2350 42844 y(circulan)-36 b(t.)-399 44450 y(On)388 b(the)g(other)h(hand,)397 b(if)389 b(switc)-36 b(hing)389 b(is)g(p)36 b(erformed)388 b(with)h(resp)36 b(ect)388 b(to)h(an)-36 b(y)389 b(other)f(non-trivial)h(set)f Fv(S)48978 43968 y Fp(0)49289 44450 y Fx(,)-2350 46055 y(this)334 b(set)f(or)h(its)g(complemen)-36 b(t)333 b(m)-36 b(ust)333 b(include)g(t)-36 b(w)g(o)334 b(consecutiv)-36 b(e)334 b(v)-36 b(ertices)334 b Fv(i)p Fx(,)354 b Fv(i)91 b Fx(+)g(1)333 b(\(mo)36 b(d)333 b Fv(k)45 b Fx(\))334 b(and)f(of)i(course)-2350 47660 y(there)452 b(exists)i Fv(j)527 b Fx(suc)-36 b(h)451 b(that)i Fv(j)476 b Fr(2)400 b Fv(S)14636 47178 y Fp(0)14947 47660 y Fx(,)458 b Fv(j)383 b Fx(+)307 b(1)402 b Fr(62)f Fv(S)21222 47178 y Fp(0)21532 47660 y Fx(.)636 b(In)452 b(a)h(circulan)-36 b(t)452 b(graph)g(either)g(1)402 b Fr(\273)f Fx(2)h Fr(\273)f(\242)221 b(\242)g(\242)402 b(\273)f Fv(k)447 b Fr(\273)401 b Fx(1)-2350 49265 y(or)495 b(1)474 b Fr(6\273)f Fx(2)g Fr(6\273)g(\242)221 b(\242)g(\242)474 b(6\273)f Fv(k)518 b Fr(6\273)474 b Fx(1;)526 b(assume,)510 b(up)494 b(to)g(complemen)-36 b(ting,)510 b(the)495 b(former.)762 b(Then)494 b(in)h(the)f(switc)-36 b(hed)-2350 50870 y(graph)434 b Fv(i)368 b Fr(\273)i Fv(i)294 b Fx(+)h(1)434 b(while)g Fv(j)444 b Fr(6\273)369 b Fv(j)h Fx(+)295 b(1;)434 b(so)g(the)f(new)h(graph)f(is)h(not)f (circulan)-36 b(t.)13370 b Fr(})-399 53139 y Fx(A)390 b(v)-72 b(ariation)390 b(of)g(the)f(previous)g(argumen)-36 b(t)389 b(sho)-36 b(ws)390 b(that)e(the)h(same)h(holds)f(for)h(orien) -36 b(ted)389 b(t)-36 b(w)g(o-graphs.)-2350 57576 y Fy(4)2152 b(Groups)717 b(and)f(their)g(sequences)-2350 60497 y Fx(In)540 b(this)g(section)g(w)-36 b(e)541 b(consider)e(the)h(tables)g (included)f(in)h(Cameron's)h(pap)36 b(er)539 b([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.Cam2000) [[409 173 415 185] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])h(and)g(add,)566 b(as)540 b(far)h(as)-2350 62102 y(p)36 b(ossible,)435 b(the)e(data)g(concerning)h(P)-36 b(ark)g(er)433 b(sequences.)-399 64371 y(F)-108 b(or)533 b(the)g(\257v)-36 b(e)533 b(closed)h(highly)f(homogeneous)h(groups)e (of)i(Cameron's)g(theorem)f(\(i.e.,)559 b(the)533 b(groups)-2350 65976 y(admitting)325 b(only)h(one)f(orbit)f(on)h Fv(k)45 b Fx(-sets)324 b(for)i(all)g Fv(k)45 b Fx(;)361 b(see)325 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.cam76) [[291 123 296 135] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\))g(the)f(P)-36 b(ark)g(er)325 b(sequences)g(are)g(readily) h(obtained.)-2350 67581 y(Recall)415 b(that)e Fv(S)490 b Fx(is)414 b(the)f(in\257nite)f(symmetric)i(group,)j Fv(A)d Fx(\(or)f Fv(@)72 b(C)95 b Fx(\))413 b(is)h(the)e(subgroup)g(of) j Fv(S)490 b Fx(of)414 b(the)f(p)36 b(erm)-36 b(u-)-2350 69186 y(tations)462 b(preserving)g(the)f(ordering)h(on)g(the)f (rational)h(n)-36 b(um)g(b)36 b(ers,)468 b Fv(B)529 b Fx(\(or)461 b Fv(@)72 b(C)35774 68704 y Fp(\244)36300 69186 y Fx(\))461 b(of)i(those)e(preserving)h(or)-2350 70792 y(rev)-36 b(ersing)445 b(it,)h Fv(C)539 b Fx(of)445 b(those)e(preserving)h(a)g(cyclic)h(order)f(on)f(a)h(coun)-36 b(table)444 b(set)f(\(sa)-36 b(y)-108 b(,)447 b(the)c(complex)i(ro)36 b(ots)-2350 72397 y(of)435 b(unit)-36 b(y\),)433 b(and)g Fv(D)470 b Fx(\(or)433 b Fv(C)10595 71915 y Fp(\244)11121 72397 y Fx(\))g(of)i(those)e(preserving)h(or)f(rev)-36 b(ersing)434 b(suc)-36 b(h)433 b(a)h(cyclic)h(order.)p Black Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black 41095 -272 a Fx(6)p Black -399 3824 a(The)617 b(P)-36 b(ark)g(er)617 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b(=)f Fu(p)p Fx(\()p Fv(G)296 b Fr(o)f Fv(A)p Fx(\))369 b(=)f Fu(p)p Fx(\()p Fv(G)p Fx(\))p Fv(:)-399 13455 y Fx(The)375 b(sequences)f(for)h Fv(C)470 b Fx(and)374 b Fv(D)410 b Fx(can)375 b(b)36 b(e)374 b(obtained)g(b)-36 b(y)374 b(noting)h(that)f(these)g(groups)g (induce)f(on)i Fv(k)45 b Fx(-sets)-2350 15060 y(the)433 b(groups)h Fv(C)5020 15259 y Fs(k)6022 15060 y Fx(and)f Fv(D)9631 15259 y Fs(k)10633 15060 y Fx(\(dihedral)g(of)h(degree)g Fv(k)45 b Fx(\),)434 b(resp)36 b(ectiv)-36 b(ely;)435 b(see)e(also)i([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.GeMe01) [[398 581 404 593] [1 1 1 [3 3]] [0 0 1]] pdfm Black(].)15384 17150 y(Highly)f(Homogeneous)g(Groups)p Black Black 10375 18770 26552 45 v 10353 20099 45 1329 v 11039 19700 a Fl(Group)p 15257 20099 V 1330 w(P)-35 b(ark)g(er)424 b(sequence)p 25637 20099 V 1330 w(EIS)h(en)-35 b(try)p 32362 20099 V 1328 w(Notes)p 36904 20099 V 10375 20143 26552 45 v 10353 21471 45 1329 v 11039 21073 a Fk(S)p 15257 21471 V 4226 w Fj(1,)369 b(1,)h(1,)g(.)185 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31760 V 10375 31804 26552 45 v 10353 33132 45 1329 v 11039 32734 a Fk(S)310 b Fi(\243)246 b Fk(S)p 15257 33132 V 2130 w Fj(2,)369 b(2,)h(2,)g(.)185 b(.)g(.)p 25637 33132 V 5399 w(A007395)p 32362 33132 V 36904 33132 V 10353 34461 V 11039 34062 a Fk(S)310 b Fi(\243)246 b Fk(A)p 15257 34461 V 1979 w Fj(2,)369 b(1,)h(1,)g(.)185 b(.)g(.)p 25637 34461 V 5399 w(A054977)p 32362 34461 V 36904 34461 V 10353 35789 V 11039 35390 a Fk(A)246 b Fi(\243)g Fk(A)p 15257 35789 V 1892 w Fj(2,)369 b(0,)h(0,)g(.)185 b(.)g(.)p 25637 35789 V 5399 w(A000038)p 32362 35789 V 36904 35789 V 10353 37117 V 11039 36719 a Fk(S)11782 36317 y Fh(3)p 15257 37117 V 15944 36719 a Fj(3,)369 b(3,)h(3,)g(.)185 b(.)g(.)p 25637 37117 V 5399 w(A010701)p 32362 37117 V 36904 37117 V 10353 38456 45 1339 v 11039 38057 a Fk(S)11782 37655 y Fg(k)p 15257 38456 V 15944 38057 a Fk(k)35 b Fj(,)369 b Fk(k)35 b Fj(,)370 b Fk(k)35 b Fj(,)369 b(.)185 b(.)g(.)p 25637 38456 V 32362 38456 V 36904 38456 V 10375 38500 26552 45 v -399 40814 a 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Fk(S)p 5280 54162 V 5343 w Fj(1,)370 b(1,)g(1,)g(.)185 b(.)g(.)p 30337 54162 V 20076 w(A000012)p 42092 54162 V 49730 54162 V -2372 55618 45 1456 v -1685 55092 a Fk(C)325 b Fi(o)246 b Fk(S)p 5280 55618 V 5303 w(k)404 b Fj(\(=)8607 54262 y Ff(P)9775 55424 y Fg(d)p Fe(j)p Fg(k)11228 55092 y Fk(')p Fj(\()p Fk(d)p Fj(\)\))p 30337 55618 V 17205 w(A000027)p 42092 55618 V 49730 55618 V -2372 57074 V -1685 56548 a(\()p Fk(C)325 b Fi(o)247 b Fk(S)64 b Fj(\))246 b Fi(o)g Fk(S)p 5280 57074 V 5966 55718 a Ff(P)7135 56880 y Fg(d)p Fe(j)p Fg(k)8588 56548 y Fk(d)307 b Fj(=)h Fk(\276)40 b Fj(\()p Fk(k)35 b Fj(\))p 30337 57074 V 18239 w(A000203)p 42092 57074 V 49730 57074 V -2372 58402 45 1329 v -1685 58003 a Fk(S)309 b Fi(o)247 b Fk(A)p 5280 58402 V 5279 w Fj(1,)370 b(1,)g(1,)g(.)185 b(.)g(.)p 30337 58402 V 20076 w(A000012)p 42092 58402 V 49730 58402 V -2372 59730 V -1685 59332 a Fk(S)309 b Fi(o)247 b Fk(S)536 59498 y Fh(2)1032 59332 y Fj(,)370 b Fk(S)2388 59498 y Fh(2)3130 59332 y Fi(o)246 b Fk(S)p 5280 59730 V 1604 w Fj(1,)370 b(2,)g(1,)g(2,)g(.)185 b(.)g(.)p 30337 59730 V 18846 w(A000034)p 42092 59730 V 49730 59730 V -2372 61059 V -1685 60660 a Fk(S)309 b Fi(o)247 b Fk(S)536 60826 y Fh(3)1032 60660 y Fj(,)370 b Fk(S)2388 60826 y Fh(3)3130 60660 y Fi(o)246 b Fk(S)p 5280 61059 V 1604 w Fj(1,)370 b(2,)g(2,)g(2,)g(1,)g(3,)g(1,)g(2,)g(2,)g(2,)g(1,)g(3,)g(.)185 b(.)g(.)p 30337 61059 V 9006 w(A083039)p 42092 61059 V 7608 w(See)369 b Fk(S)309 b Fi(o)247 b Fk(S)46968 60826 y Fg(n)p 49730 61059 V -2372 62387 V -1685 61988 a Fk(S)309 b Fi(o)247 b Fk(S)536 62154 y Fh(4)1032 61988 y Fj(,)370 b Fk(S)2388 62154 y Fh(4)3130 61988 y Fi(o)246 b Fk(S)p 5280 62387 V 1604 w Fj(1,)370 b(2,)g(2,)g(3,)g(1,)g(3,)g(1,)g(3,)g(2,)g (2,)g(1,)g(4,)g(.)185 b(.)g(.)p 30337 62387 V 9006 w(A083040)p 42092 62387 V 7608 w(See)369 b Fk(S)309 b Fi(o)247 b Fk(S)46968 62154 y Fg(n)p 49730 62387 V -2372 63715 V -1685 63317 a Fk(S)309 b Fi(o)247 b Fk(S)536 63483 y Fg(n)1138 63317 y Fj(,)369 b Fk(S)2493 63483 y Fg(n)3342 63317 y Fi(o)246 b Fk(S)p 5280 63715 V 1392 w(p)6523 63483 y Fg(k)7375 63317 y Fj(=)308 b Fi(jf)p Fk(d)g Fj(:)g Fk(d)p Fi(j)p Fk(k)35 b(;)371 b(d)307 b Fi(\267)h Fk(n)p Fi(gj)p 30337 63715 V 42092 63715 V 26128 w Fj(See)369 b(remark)g(1)p 49730 63715 V -2372 65171 45 1456 v -1685 64645 a Fk(S)309 b Fi(o)247 b Fk(S)309 b Fi(o)247 b Fk(S)p 5280 65171 V 5966 63815 a Ff(P)7135 64977 y Fg(d)7596 65088 y Fd(0)8028 64977 y Fe(j)p Fg(k)9020 64645 y Fk(d)p Fj(\()p Fk(d)10602 64811 y Fh(0)11099 64645 y Fj(\))308 b(=)f(1)p Fk(;)184 b Fj(3)p Fk(;)g Fj(3)p Fk(;)g Fj(6)p Fk(;)g Fj(3)p Fk(;)g Fj(9)p Fk(;)g Fj(3)p Fk(;)g Fj(10)p Fk(;)g Fj(6)p Fk(;)g Fj(9)p Fk(;)g Fj(3)p Fk(;)g(:)g(:)g(:)p 30337 65171 V 4692 w Fj(A007425)p 42092 65171 V 49730 65171 V -2372 66499 45 1329 v -1685 66101 a Fk(A)246 b Fi(o)g Fk(A)p 5280 66499 V 5192 w Fj(1,)370 b(0,)g(0,)g(.)185 b(.)g(.)p 30337 66499 V 20076 w(A000007)p 42092 66499 V 49730 66499 V -2372 67828 V -1685 67429 a Fk(S)-1006 67595 y Fg(k)-216 67429 y Fi(o)247 b Fk(A)p 5280 67828 V 4798 w Fj(1,)370 b(.)185 b(.)g(.)g(\()p Fk(k)405 b Fj(times\))370 b(.)185 b(.)g(.)g(,)370 b(1,)g(0,)g(0,)g(.)185 b(.)g(.)p 30337 67828 V 42092 67828 V 49730 67828 V -2372 69156 V -1685 68758 a Fk(E)310 b Fi(o)246 b Fk(S)p 5280 69156 V 5292 w Fj(2,)370 b(2,)g(2,)g(.)185 b(.)g(.)p 30337 69156 V 20076 w(A007395)p 42092 69156 V 49730 69156 V -2372 70484 V -1685 70086 a Fk(E)310 b Fi(o)246 b Fk(A)p 5280 70484 V 5141 w Fj(2,)370 b(0,)g(0,)g(.)185 b(.)g(.)p 30337 70484 V 20076 w(A000038)p 42092 70484 V 49730 70484 V -2372 71959 45 1475 v -1685 71424 a Fk(S)-942 71022 y Fe(o)p Fg(n)p 5280 71959 V 5966 70594 a Ff(P)7135 71756 y Fg(d)7596 71867 y Fd(0)8028 71756 y Fe(j)p Fg(k)9020 70594 y Ff(P)10188 71756 y Fg(d)10649 71867 y Fd(1)11081 71756 y Fe(j)p Fg(d)11805 71867 y Fd(0)12476 70594 y Ff(P)13645 71756 y Fg(d)14106 71867 y Fd(2)14538 71756 y Fe(j)p Fg(d)15262 71867 y Fd(1)15933 71424 y Fi(\242)f(\242)g(\242) 17409 70594 y Ff(P)18578 71756 y Fg(d)19039 71867 y Fc(n)p Fb(\241)p Fd(3)20558 71756 y Fe(j)p Fg(d)21282 71867 y Fc(n)p Fb(\241)p Fd(4)23040 71424 y Fk(d)p Fj(\()p Fk(d)24622 71590 y Fg(n)p Fe(\241)p Fh(3)26359 71424 y Fj(\))p 30337 71959 V 4234 w(MOBIUSi)35835 71022 y Fg(n)36438 71424 y Fj(\(A000005\))p 42092 71959 V 1333 w(See)369 b(remark)g(2)p 49730 71959 V -2372 73415 45 1456 v -1685 72889 a Fk(C)325 b Fi(o)246 b Fk(C)p 5280 73415 V 5966 72059 a Ff(P)7135 73221 y Fg(d)p Fe(j)p Fg(k)8588 72889 y Fk(')p Fj(\()p Fk(d)p Fj(\))p Fk(')p Fj(\()p Fk(k)35 b(=d)p Fj(\))310 b(=)e(1)p Fk(;)184 b Fj(2)p Fk(;)g Fj(4)p Fk(;)g Fj(5)p Fk(;)g Fj(8)p Fk(;)g Fj(8)p Fk(;)g Fj(12)p Fk(;)g Fj(12)p Fk(;)g Fj(16)p Fk(;)g Fj(16)p Fk(;)g(:)g(:)g(:)p 30337 73415 V 1531 w Fj(A029935)p 42092 73415 V 7608 w(See)369 b(remark)g(3)p 49730 73415 V -2350 73459 52103 45 v Black Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black 41095 -272 a Fx(7)p Black -2350 3824 a Fu(Remark)499 b(1)1301 b Fx(The)433 b(sequence)h(is)g(p)36 b(erio)g(dic)434 b(of)g(p)36 b(erio)g(d)434 b(lcm\(1)p Fv(;)221 b(:)g(:)g(:)j(;)d(n)p Fx(\).)-2350 5761 y Fu(Remark)524 b(2)1301 b Fx(The)455 b(sequence)g(asso)36 b(ciated)457 b(with)e Fv(S)23572 5279 y Fp(o)p Fs(n)24915 5761 y Fx(\(i.e.,)462 b(the)454 b(iterated)h(wreath)h(pro)36 b(duct)454 b(of)i Fv(S)532 b Fx(with)-2350 7366 y(itself)407 b(with)f Fv(n)g Fx(factors\))g(can)g(b)36 b(e)405 b(expressed)h(as)g (follo)-36 b(ws.)571 b(Let)405 b Fv(\261)28719 7565 y Ft(0)29245 7366 y Fx(\()p Fv(k)45 b Fx(\))368 b(:=)h(1)406 b(for)g(eac)-36 b(h)406 b Fv(k)45 b Fx(,)412 b(and)405 b(for)h Fv(i)369 b(>)f Fx(0)406 b(let)18081 10532 y Fv(\261)18657 10731 y Fs(i)19033 10532 y Fx(\()p Fv(k)45 b Fx(\))369 b(:=)22877 9270 y Fq(X)23206 12148 y Fs(d)p Fp(j)p Fs(k)25017 10532 y Fv(\261)25593 10731 y Fs(i)p Fp(\241)p Ft(1)27171 10532 y Fx(\()p Fv(d)p Fx(\))p Fv(;)-2350 15039 y Fx(that)434 b(is,)g Fv(\261)2714 15238 y Fs(i)3523 15039 y Fx(is)g(the)f(Diric)-36 b(hlet)434 b(con)-36 b(v)g(olution)434 b Fv(\261)19985 15238 y Fs(i)p Fp(\241)p Ft(1)21858 15039 y Fr(\244)296 b Fv(\261)23394 15238 y Ft(0)23919 15039 y Fx(.)579 b(Th)-36 b(us,)433 b Fv(\261)29091 15238 y Fs(i)29467 15039 y Fx(\()p Fv(k)45 b Fx(\))369 b(=)f Fv(p)33602 15238 y Fs(k)34171 15039 y Fx(\()p Fv(S)35554 14557 y Fp(o)p Fs(i)p Ft(+1)37393 15039 y Fx(\).)-399 16644 y(All)478 b(the)f(functions)g Fv(\261)10287 16843 y Fs(i)11140 16644 y Fx(are)h(m)-36 b(ultiplicativ)g(e,)489 b(b)36 b(ecause)477 b Fv(\261)27401 16843 y Ft(0)28404 16644 y Fx(is,)489 b(and)477 b(the)g(Diric)-36 b(hlet)478 b(con)-36 b(v)g(olution)477 b(pre-)-2350 18249 y(serv)-36 b(es)434 b(m)-36 b(ultiplicativit)g(y)-108 b(.)580 b(Th)-36 b(us,)433 b(it)h(su\261ces)f(to)h(compute)f(the)g(v)-72 b(alue)434 b(of)g Fv(\261)34898 18448 y Fs(i)35707 18249 y Fx(on)g(prime)f(p)36 b(o)-36 b(w)g(ers.)-399 19854 y(W)-108 b(e)434 b(claim)g(that)18702 21902 y Fv(\261)19278 22101 y Fs(i)19654 21902 y Fx(\()p Fv(p)20813 21353 y Fs(j)21299 21902 y Fx(\))369 b(=)23555 20029 y Fq(\265)24456 21088 y Fv(i)295 b Fx(+)g Fv(j)25564 22693 y(i)27038 20029 y Fq(\266)28237 21902 y Fv(:)-399 25001 y Fx(T)-108 b(o)454 b(obtain)g(a)g(di\256eren)-36 b(t)452 b(description)h(of)i(the)e Fv(\261)22870 25200 y Fs(i)23246 25001 y Fx(s,)459 b(note)453 b(that)g Fv(\261)30903 25200 y Ft(1)31429 25001 y Fx(\()p Fv(k)45 b Fx(\))453 b(giv)-36 b(es)455 b(the)e(n)-36 b(um)g(b)36 b(er)452 b(of)j(divisors)-2350 26607 y(of)391 b Fv(k)45 b Fx(,)399 b(including)390 b(1)g(and)f Fv(k)45 b Fx(;)405 b(so)390 b(it)g(is)g(equal)h(to)f Fv(d)p Fx(\()p Fv(k)45 b Fx(\).)563 b(Next,)399 b Fv(\261)27597 26806 y Ft(2)28123 26607 y Fx(\()p Fv(k)45 b Fx(\))390 b(is)g(the)f(sum)h(o)-36 b(v)g(er)390 b(the)f(divisors)i(of)g Fv(k)435 b Fx(of)-2350 28212 y(the)401 b(n)-36 b(um)g(b)36 b(er)399 b(of)i(their)g(divisors;) 413 b(in)400 b(other)h(w)-36 b(ords,)407 b(it)401 b(giv)-36 b(es)402 b(the)e(n)-36 b(um)g(b)36 b(er)400 b(of)h(pairs)g(\()p Fv(h;)221 b(d)p Fx(\))401 b(with)g Fv(h)p Fr(j)p Fv(d)f Fx(and)-2350 29817 y Fv(d)p Fr(j)p Fv(k)443 b Fx(\(observ)-36 b(e)398 b(that)g Fv(h)f Fx(and)h Fv(d)f Fx(ma)-36 b(y)399 b(w)-36 b(ell)399 b(coincide\).)566 b(In)398 b(general,)406 b(w)-36 b(e)398 b(see)g(that)f Fv(\261)37489 30016 y Fs(i)37865 29817 y Fx(\()p Fv(k)45 b Fx(\))398 b(giv)-36 b(es)399 b(the)e(n)-36 b(um)g(b)36 b(er)-2350 31422 y(of)482 b Fv(i)p Fx(-ples)d(\()p Fv(d)3893 31621 y Ft(1)4419 31422 y Fv(;)221 b(d)5677 31621 y Ft(2)6203 31422 y Fv(;)g(:)g(:)g(:)j (;)d(d)9792 31621 y Fs(i)10168 31422 y Fx(\))480 b(with)h Fv(d)14840 31621 y Ft(1)15365 31422 y Fr(j)p Fv(d)16410 31621 y Ft(2)16936 31422 y Fx(,)492 b(.)217 b(.)g(.)g(,)493 b Fv(d)21053 31621 y Fs(i)p Fp(\241)p Ft(1)22631 31422 y Fr(j)p Fv(d)23676 31621 y Fs(i)24051 31422 y Fx(,)g Fv(d)25581 31621 y Fs(i)25956 31422 y Fr(j)p Fv(k)45 b Fx(.)719 b(W)-108 b(e)481 b(call)g(suc)-36 b(h)480 b(a)g(sequence)h(a)f Fw(gener)-66 b(alise)g(d)-2350 33027 y(gozinta)499 b(chain)p Fx(,)479 b(recalling)472 b(that)f(a)g(gozin)-36 b(ta)471 b(\(\\go)36 b(es)472 b(in)-36 b(to"\))471 b(c)-36 b(hain)470 b(for)i Fv(k)516 b Fx(is)471 b(a)g(sequence)g(of)g(divisors) h(of)-2350 34632 y Fv(k)479 b Fx(eac)-36 b(h)434 b(of)g(whic)-36 b(h)433 b(strictly)h(divides)g(the)f(next)h(one.)-399 36237 y(When)317 b Fv(k)414 b Fx(=)369 b Fv(p)6402 35755 y Fs(j)6889 36237 y Fx(,)341 b(a)317 b(sequence)h(of)g(divisors)g(of)g Fv(k)363 b Fx(eac)-36 b(h)317 b(of)h(whic)-36 b(h)317 b(divides)h(the)f(next)g(one)h(corresp)36 b(onds)316 b(to)-2350 37842 y(a)415 b(nondecreasing)g(sequence)g(of)g(exp)36 b(onen)-36 b(ts)414 b(of)i Fv(p)p Fx(,)i(that)d(is)g(to)f(a)h (nondecreasing)g(sequence)g(of)g(n)-36 b(um)g(b)36 b(ers)-2350 39447 y(in)434 b([)p Fv(j)75 b Fx(])369 b(=)g Fr(f)p Fx(0)p Fv(;)221 b Fx(1)p Fv(;)g(:)g(:)g(:)k(;)c(j)75 b Fr(g)p Fx(,)435 b(whic)-36 b(h)433 b(in)g(turn)g(can)g(b)36 b(e)434 b(seen)f(as)h(a)g(m)-36 b(ultiset)433 b(of)h(elemen)-36 b(ts)434 b(of)g([)p Fv(j)75 b Fx(].)-399 41052 y(So,)442 b(it)f(is)f(enough)g(to)g(en)-36 b(umerate)439 b(the)h(m)-36 b(ultisubsets)439 b(of)i Fr(f)p Fx(0)p Fv(;)221 b Fx(1)p Fv(;)g(:)g(:)g(:)k(;)c(j)75 b Fr(g)441 b Fx(of)f(size)h Fv(i)p Fx(.)598 b(It)440 b(is)g(w)-36 b(ell)441 b(kno)-36 b(wn)-2350 43432 y(\(see)434 b(for)g(instance)f([)p 0 1 0 0 TeXcolorcmyk(14)p (#cite.stanley) [[141 326 153 338] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\))h(that)f(their)g(n)-36 b(um)g(b)36 b(er)432 b(is)i(giv)-36 b(en)434 b(b)-36 b(y)27443 41559 y Fq(\265)28344 42619 y Fv(i)295 b Fx(+)g Fv(j)29452 44224 y(i)30926 41559 y Fq(\266)31904 43432 y Fx(,)434 b(as)g(claimed.)-2350 46144 y Fu(Remark)499 b(3)1301 b Fx(If)434 b Fv(k)479 b Fx(is)433 b(square-free,)i Fv(p)16724 46343 y Fs(k)17726 46144 y Fx(is)f(equal)g(to)24056 45148 y Fq(P)25458 46532 y Fs(d)p Fp(j)p Fs(k)26994 46144 y Fv(')p Fx(\()p Fv(k)45 b Fx(\))368 b(=)h Fv(d)p Fx(\()p Fv(k)45 b Fx(\))p Fv(')p Fx(\()p Fv(k)g Fx(\).)p Black Black 20074 51956 a(*)2601 b(*)h(*)-399 57657 y(The)434 b(follo)-36 b(wing)436 b(groups)d(arise)h (as)g(automorphism)f(groups)g(of)i(F)-108 b(ra)-145 b(\304)-506 b(\263ss)-36 b(\266)-614 b(e)434 b(classes)h(\(see)e(section)p 0 1 0 0 TeXcolorcmyk 434 w(3)p (#section.3) [[488 198 494 210] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)-399 59262 y(The)333 b(calculation)h(of)g(P)-36 b(ark)g(er)333 b(sequences)g(for)g(\\treelik)-36 b(e)334 b(ob)72 b(jects")334 b(and)f(related)g(structures)e(is)i(carried)-2350 60867 y(out)434 b(in)f(detail)h(in)g(the)f(forthcoming)h(pap)36 b(er)433 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[253 169 259 181] [1 1 1 [3 3]] [0 0 1]] pdfm Black(].)-399 62472 y(The)h(letters)g(R)f(and)g(L)h(mean)f(\\shifted)h (righ)-36 b(t")433 b(and)g(\\shifted)h(left")h(resp)36 b(ectiv)-36 b(ely)-108 b(.)p Black Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black 41095 -272 a Fx(8)p Black 9126 3824 a(Automorphism)433 b(Groups)f(of)j(Homogeneous)f (Structures)p Black Black -1945 6341 51191 45 v -1967 7669 45 1329 v -1281 7270 a Fl(F)-106 b(ra)-141 b(\304)-495 b(\263ss)-27 b(\266)-610 b(e)426 b(class)p 15109 7669 V 10330 w(P)-35 b(ark)g(er)424 b(sequence)p 34390 7669 V 10232 w(EIS)g(en)-35 b(try)p 41500 7669 V 1713 w(Notes)p 49224 7669 V -1945 7713 51191 45 v -1967 9042 45 1329 v 15109 9042 V 34390 9042 V 41500 9042 V 49224 9042 V -1967 10423 45 1382 v -1281 10024 a Fj(Graphs)p 15109 10423 V 13554 w(2)16349 9623 y Fe(b)p Fg(k)19 b(=)p Fh(2)p Fe(c)p 34390 10423 V 35077 10024 a Fj(A016116)p 41500 10423 V 2961 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.GeMe01) [[472 627 477 639] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 10423 V -1967 11804 V -1281 11406 a(Graphs)g(up)f(to)i(complemen)-31 b(t)p 15109 11804 V 4425 w Fk(p)16353 11572 y Fh(1)17157 11406 y Fj(=)307 b(1,)370 b Fk(p)20112 11572 y Fg(k)20964 11406 y Fj(=)308 b(2)22686 11004 y Fe(b)p Fg(k)19 b(=)p Fh(2)p Fe(c\241)p Fh(1)26439 11406 y Fj(for)370 b Fk(k)342 b(>)308 b Fj(1)p 34390 11804 V 4303 w(A016116RR)p 41500 11804 V 1331 w(See)369 b(rem.)g(4)p 49224 11804 V -1967 13133 45 1329 v -1281 12734 a Fk(K)-341 12900 y Fh(3)156 12734 y Fj(-free)g(graphs)p 15109 13133 V 9939 w(1,2,1,3,3,4,4,8,4,14,11,14,.)198 b(.)185 b(.)p 34390 13133 V 5998 w(A083041)p 41500 13133 V 2961 w(See)369 b(rem.)g(5)p 49224 13133 V -1967 14461 V -1281 14063 a(Graphs)g(with)h(bipartite)h(blo)31 b(c)-31 b(k)p 15109 14461 V 3559 w(2,2,2,.)188 b(.)c(.)p 34390 14461 V 15408 w(A007395)p 41500 14461 V 2961 w(See)369 b(rem.)g(6)p 49224 14461 V -1967 15842 45 1382 v -1281 15444 a(Graphs)g(with)h(lo)31 b(ops)p 15109 15842 V 8168 w(2)16349 15042 y Fe(b)p Fg(k)19 b(=)p Fh(2)p Fe(c)p Fh(+1)p 34390 15842 V 35077 15444 a Fj(A016116LL)p 41500 15842 V 1577 w(See)369 b(rem.)g(7)p 49224 15842 V -1967 17224 V -1281 16825 a(Digraphs)p 15109 17224 V 12717 w(2)16349 16424 y Fg(k)19 b Fe(\241)p Fh(1)18027 16825 y Fj(\(=)308 b(4)20179 16424 y Fh(\()p Fg(k)19 b Fe(\241)p Fh(1\))p Fg(=)p Fh(2)23444 16825 y Fj(\))p 34390 17224 V 11203 w(A000079R)p 41500 17224 V 49224 17224 V -1967 19890 45 2667 v -1281 18164 a(Digraphs)554 b(with)g(lo)31 b(ops)554 b(\(or)g(binary)-1281 19492 y(relations\))p 15109 19890 V 15796 18164 a(2)16349 17762 y Fg(k)p 34390 19890 V 35077 18164 a Fj(A000079)p 41500 19890 V 49224 19890 V -1967 21272 45 1382 v -1281 20873 a(Orien)-31 b(ted)370 b(graphs)p 15109 21272 V 9285 w(3)16349 20472 y Fe(b)p Fg(k)19 b(=)p Fh(2)p Fe(c)p 34390 21272 V 35077 20873 a Fj([missing])p 41500 21272 V 49224 21272 V -1967 22600 45 1329 v -1281 22202 a(T)-92 b(op)31 b(ologies)p 15109 22600 V 11969 w Fk(d)p Fj(\()p Fk(k)k Fj(\))p 34390 22600 V 17234 w(A000005)p 41500 22600 V 2961 w(See)369 b(rem.)g(8)p 49224 22600 V -1967 23929 V -1281 23530 a(P)-31 b(osets)p 15109 23929 V 14006 w(1,1,1,.)188 b(.)c(.)p 34390 23929 V 15408 w(A000012)p 41500 23929 V 2961 w(See)369 b(rem.)g(9)p 49224 23929 V -1967 25310 45 1382 v -1281 24911 a(T)-92 b(ournamen)-31 b(ts)p 15109 25310 V 10735 w Fk(k)404 b Fj(o)31 b(dd:)492 b(2)19942 24510 y Fe(b)p Fg(k)19 b(=)p Fh(2)p Fe(c)22194 24911 y Fj(,)370 b Fk(k)404 b Fj(ev)-31 b(en:)493 b(0)p 34390 25310 V 7721 w([missing])p 41500 25310 V 2917 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.GeMe01) [[472 493 477 505] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 25310 V -1967 26638 45 1329 v -1281 26240 a(Lo)31 b(cal)370 b(orders)p 15109 26638 V 11114 w Fk(k)404 b Fj(o)31 b(dd:)492 b Fk(')p Fj(\()p Fk(k)35 b Fj(\),)371 b Fk(k)404 b Fj(ev)-31 b(en:)493 b(0)p 34390 26638 V 8330 w([missing])p 41500 26638 V 2917 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 481 477 493] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 26638 V -1967 28020 45 1382 v -1281 27621 a(Tw)-31 b(o-graphs)p 15109 28020 V 11381 w(2)16349 27219 y Fe(d)p Fg(k)19 b(=)p Fh(2)p Fe(e)p 34390 28020 V 35077 27621 a Fj(A016116L)p 41500 28020 V 2269 w(See)369 b(Thm.)p 0 1 0 0 TeXcolorcmyk 370 w(3.1)p (#tm.3.1) [[497 468 509 480] [1 1 1 [3 3]] [0 0 1]] pdfm Black 49224 28020 V -1967 29401 V -1281 29002 a(Orien)-31 b(ted)370 b(t)-31 b(w)g(o-graphs)p 15109 29401 V 7196 w(2)16349 28601 y Fe(d)p Fg(k)19 b(=)p Fh(2)p Fe(e)p 34390 29401 V 35077 29002 a Fj(A016116L)p 41500 29401 V 2269 w(See)369 b(Thm.)p 0 1 0 0 TeXcolorcmyk 370 w(3.1)p (#tm.3.1) [[497 456 509 468] [1 1 1 [3 3]] [0 0 1]] pdfm Black 49224 29401 V -1967 30729 45 1329 v -1281 30331 a(T)-92 b(otal)371 b(orders)e(with)h(subset)p 15109 30729 V 5275 w(2,0,0,.)188 b(.)c(.)p 34390 30729 V 15408 w(A000038)p 41500 30729 V 49224 30729 V -1967 32058 V -1281 31659 a(T)-92 b(otal)371 b(orders)e(with)h(2-partition)p 15109 32058 V 3135 w(1,0,0,.)188 b(.)c(.)p 34390 32058 V 15408 w(A000007)p 41500 32058 V 49224 32058 V -1967 33386 V -1281 32988 a Fk(C)79 b Fj(-structures)369 b(with)h(subset)p 15109 33386 V 5106 w(2)p Fk(')p Fj(\()p Fk(k)35 b Fj(\))p 34390 33386 V 16533 w([missing])p 41500 33386 V 2917 w(See)369 b(rem.)g(10)p 49224 33386 V -1967 34714 V -1281 34316 a Fk(D)31 b Fj(-structures)369 b(with)h(subset)p 15109 34714 V 5029 w Fk(')p Fj(\()p Fk(k)35 b Fj(\))p 34390 34714 V 17086 w(A000010)p 41500 34714 V 2961 w(See)369 b(rem.)g(10)p 49224 34714 V -1967 36043 V -1281 35644 a(2)h(total)h(orders)d(\(distinguished\))p 15109 36043 V 2972 w(1,0,0,.)188 b(.)c(.)p 34390 36043 V 15408 w(A000007)p 41500 36043 V 49224 36043 V -1967 37371 V -1281 36973 a(2)288 b(total)h(orders)e(\(not)h(distinguished\))p 15109 37371 V 1331 w(1,1,0,0,.)188 b(.)d(.)p 34390 37371 V 14547 w(A019590)p 41500 37371 V 49224 37371 V -1967 40028 45 2657 v -1281 38301 a(2)1091 b(b)31 b(et)-31 b(w)g(eennesses)1090 b(\(not)h(distin-)-1281 39629 y(guished\))p 15109 40028 V 15796 38301 a(1,1,0,0,.)188 b(.)d(.)p 34390 40028 V 14547 w(A019590)p 41500 40028 V 49224 40028 V -1967 41366 45 1339 v -1281 40968 a(Boron)370 b(trees)e(\(lea)-31 b(v)g(es\))372 b(\(or)d Fk(T)11149 41134 y Fh(3)11646 40968 y Fj(\))p 15109 41366 V 3720 w(c)-31 b(haracteristic)357 b(fn.)488 b(of)355 b Fi(f)p Fj(3)26690 40566 y Fg(a)27227 40968 y Fj(2)27780 40566 y Fg(b)28225 40968 y Fi(g)28778 41168 y Fg(a)p Fe(2f)p Fh(0)p Fg(;)p Fh(1)p Fe(g)p Fg(;b)p Fe(\270)p Fh(0)p 34390 41366 V 35077 40968 a Fj([missing])p 41500 41366 V 2917 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 348 477 360] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 41366 V -1967 42694 45 1329 v -1281 42296 a(HI)g(trees)g(\(lea)-31 b(v)g(es\))371 b(\(or)f Fk(T)154 b Fj(\))p 15109 42694 V 5772 w(nr.)492 b(of)369 b(ordered)g(factorisations)j(of)d Fk(k)p 34390 42694 V 3522 w Fj(A002033R)p 41500 42694 V 2146 w(See)g([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 336 477 348] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 42694 V -1967 44023 V -1281 43624 a(R\(Boron)h(trees)f(\(lea)-31 b(v)g(es\)\))372 b(\(or)d Fk(@)61 b(T)13474 43790 y Fh(3)13971 43624 y Fj(\))p 15109 44023 V 1395 w(c)-31 b(haracteristic)370 b(fn.)493 b(of)370 b(p)31 b(o)-31 b(w)g(ers)369 b(of)h(2)p 34390 44023 V 3978 w(A036987)p 41500 44023 V 2961 w(See)f([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 324 477 336] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 44023 V -1967 45351 V -1281 44953 a(R\(HI)h(trees)e(\(lea)-31 b(v)g(es\)\))372 b(\(or)d Fk(@)61 b(T)154 b Fj(\))p 15109 45351 V 3448 w(nr.)492 b(of)369 b(ordered)g(factorisations)j(of)d Fk(k)p 34390 45351 V 3522 w Fj(A002033R)p 41500 45351 V 2146 w(See)g([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 312 477 324] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 45351 V -1967 46680 V -1281 46281 a(T)-92 b(rees)369 b(\(edges\))p 15109 46680 V 10697 w(1,1,1,.)188 b(.)c(.)p 34390 46680 V 15408 w(A000012)p 41500 46680 V 2961 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 300 477 312] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 46680 V -1967 48008 V -1281 47609 a(Co)-31 b(vington)302 b(structures)c(\(or)i Fk(@)61 b(T)12126 47775 y Fh(3)12622 47609 y Fj(\(2\)\))p 15109 48008 V 1331 w Fk(p)16353 47818 y Fh(2)16794 47597 y Fc(i)17507 47609 y Fj(=)308 b(2)19229 47208 y Fg(i)19598 47609 y Fj(,)369 b(0)h(otherwise)p 34390 48008 V 9321 w(A048298)p 41500 48008 V 2961 w(See)f([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 289 477 301] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 48008 V -1967 49336 V -1281 48938 a(Binary)h(trees)f(\(or)g Fk(@)61 b(P)154 b(T)8967 49104 y Fh(3)9463 48938 y Fj(\))p 15109 49336 V 5903 w(1,0,0,0,.)188 b(.)d(.)p 34390 49336 V 14547 w(A000007)p 41500 49336 V 2961 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 277 477 289] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 49336 V -1967 51993 45 2657 v -1281 50266 a(Binary)426 b(trees)e(up)h(to)g(re\260ection)h(\(or)-1281 51594 y Fk(@)61 b(P)233 51193 y Fe(\244)742 51594 y Fk(T)1389 51760 y Fh(3)1885 51594 y Fj(\))p 15109 51993 V 15796 50266 a(1,1,0,0,.)188 b(.)d(.)p 34390 51993 V 14547 w(A019590)p 41500 51993 V 2961 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 265 477 277] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 51993 V -1967 53321 45 1329 v -1281 52923 a(Plane)i(trees)d(\(or)i Fk(P)154 b(T)g Fj(\))p Fi(y)p 15109 53321 V 6958 w Fk(')p Fj(\()p Fk(k)35 b Fj(\))p 34390 53321 V 17086 w(A000010)p 41500 53321 V 2961 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 241 477 253] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 53321 V -1967 55978 45 2657 v -1281 54251 a(Plane)537 b(trees)f(up)g(to)h(re\260ection)g(\(or)-1281 55579 y Fk(P)-416 55178 y Fe(\244)93 55579 y Fk(T)154 b Fj(\))p Fi(y)p 15109 55978 V 15796 54251 a(\273)307 b Fk(')p Fj(\()p Fk(k)35 b Fj(\))p Fk(=)p Fj(2)p 34390 55978 V 14812 w(A023022)p 41500 55978 V 2961 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 229 477 241] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 55978 V -1967 57306 45 1329 v -1281 56908 a(Plane)i(b)31 b(oron)369 b(trees)f(\(or)i Fk(P)154 b(T)10932 57074 y Fh(3)11428 56908 y Fj(\))p 15109 57306 V 3938 w(1,1,2,0,0,.)189 b(.)c(.)p 34390 57306 V 13686 w([missing])p 41500 57306 V 2917 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 205 477 217] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 57306 V -1967 59963 45 2657 v -1281 58236 a(Plane)261 b(b)31 b(oron)259 b(trees)g(up)g(to)h(re\260ection) -1281 59565 y(\(or)370 b Fk(P)1371 59163 y Fe(\244)1879 59565 y Fk(T)2526 59731 y Fh(3)3023 59565 y Fj(\))p 15109 59963 V 15796 58236 a(1,1,1,0,0,.)189 b(.)c(.)p 34390 59963 V 13686 w([missing])p 41500 59963 V 2917 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.GeMe??) [[472 193 477 205] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 59963 V -1967 62673 45 2710 v -1281 60946 a(3-h)-31 b(yp)31 b(ergraphs)p 15109 62673 V 10208 w(2)16349 60544 y Fg(f)87 b Fh(\()p Fg(k)19 b(;)p Fh(3\))18808 60946 y Fj(,)2513 b(where)2084 b Fk(f)119 b Fj(\()p Fk(k)35 b(;)184 b Fj(3\))3167 b(=)15796 62274 y(0)p Fk(;)184 b Fj(0)p Fk(;)g Fj(1)p Fk(;)g Fj(1)p Fk(;)g Fj(4)p Fk(;)g Fj(4)p Fk(;)g Fj(5)p Fk(;)g Fj(7)p Fk(;)g Fj(10)p Fk(;)g Fj(12)p Fk(;)g Fj(15)p Fk(;)g Fj(19)p Fk(;)g(:)g(:)g(:)p 34390 62673 V 35077 60946 a Fj([missing])p 41500 62673 V 2917 w(See)369 b([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.GeMe01) [[472 168 477 180] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 62673 V -1967 64054 45 1382 v -1281 63656 a Fk(t)p Fj(-h)-31 b(yp)31 b(ergraphs)p Fi(y)p 15109 64054 V 9869 w Fj(2)16349 63254 y Fg(f)87 b Fh(\()p Fg(k)19 b(;t)p Fh(\))p 34390 64054 V 41500 64054 V 42186 63656 a Fj(See)369 b([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.GeMe01) [[472 144 477 156] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])p 49224 64054 V -1967 65545 45 1491 v -1281 65147 a(T)-92 b(ernary)369 b(relations)p 15109 65545 V 8761 w(2)16349 64745 y Fg(k)16839 64411 y Fd(2)p 34390 65545 V 35077 65147 a Fj(A002416)p 41500 65545 V 49224 65545 V -1967 67036 V -1281 66638 a(Quaternary)h(relations)p 15109 67036 V 7008 w(2)16349 66236 y Fg(k)16839 65902 y Fd(3)p 34390 67036 V 35077 66638 a Fj([missing])p 41500 67036 V 49224 67036 V -1945 67080 51191 45 v -1281 69339 a Fi(y)f Fj(Not)h(in)g([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.Cam2000) [[102 93 107 105] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)p Black Black eop %%Page: 9 9 9 8 bop Black 0 TeXcolorgray Black 41095 -272 a Fx(9)p Black -2350 3824 a Fu(Remark)650 b(4)1301 b Fx(Eac)-36 b(h)564 b(\(sym)-36 b(b)36 b(ol)566 b(for)f(a\))g(circulan)-36 b(t)565 b(graph)g(represen)-36 b(ts)563 b(also)j(its)f(complemen)-36 b(t,)598 b(so)565 b(\(for)-2350 5429 y Fv(k)595 b(>)548 b Fx(1\))540 b(eac)-36 b(h)539 b(term)g(is)h(one)f(half)h(of)g(the)f (corresp)36 b(onding)539 b(term)g(for)h(graphs.)896 b(F)-108 b(or)539 b(instance,)566 b Fv(p)46365 5628 y Ft(2)47439 5429 y Fx(=)549 b(1)-2350 7034 y(b)36 b(ecause)434 b(the)f(graphs)g Fv(K)10008 7233 y Ft(2)10967 7034 y Fx(and)g Fv(N)14538 7233 y Ft(2)15497 7034 y Fx(are)h(no)-36 b(w)434 b(iden)-36 b(ti\257ed.)-2350 9304 y Fu(Remark)599 b(5)1300 b Fx(This)521 b(is)g(the)f(n)-36 b(um)g(b)36 b(er)518 b(of)j(symmetric)g(sum-free)f (subsets)f(of)j Fu(Z)p Fv(=)p Fx(\()p Fv(k)45 b Fx(\))39832 8822 y Fp(\244)40878 9304 y Fx(\(see)520 b([)p 0 1 0 0 TeXcolorcmyk(3)p (#cite.cam87) [[467 633 473 645] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\):)752 b(if)521 b(the)-2350 10909 y(sym)-36 b(b)36 b(ol)401 b(con)-36 b(tains)399 b Fv(a)h Fx(and)f Fv(b)p Fx(,)407 b(it)400 b(cannot)f(con)-36 b(tain)400 b Fv(a)226 b Fx(+)f Fv(b)p Fx(,)407 b(and)399 b(\(as)h(for)g(generic)g (graphs\))f(if)i(it)f(con)-36 b(tains)399 b Fv(a)p Fx(,)-2350 12514 y(it)434 b(m)-36 b(ust)433 b(con)-36 b(tain)433 b Fv(k)341 b Fr(\241)295 b Fv(a)p Fx(.)-2350 14783 y Fu(Remark)552 b(6)1301 b Fx(W)-108 b(e)480 b(cannot)g(exc)-36 b(hange)481 b(\\blac)-36 b(k")481 b(and)f(\\white")h(v)-36 b(ertices,)492 b(so)481 b(a)f(circulan)-36 b(t)480 b(structure)f(is) -2350 16388 y(an)434 b(all-blac)-36 b(k)435 b(or)e(all-white)h(n)-36 b(ull)434 b(graph.)-2350 18657 y Fu(Remark)484 b(7)1301 b Fx(Reason)422 b(as)f(in)f(section)p 0 1 0 0 TeXcolorcmyk 421 w(3)p (#section.3) [[227 549 233 561] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)k(but)c(tak)-36 b(e)422 b(in)e(addition)h(to)g(\\basic")h (circulan)-36 b(t)420 b(graphs)h(\(those)-2350 20262 y(with)409 b(sym)-36 b(b)36 b(ol)409 b(of)g(the)f(form)h Fr(f)p Fv(i;)221 b(k)290 b Fr(\241)244 b Fv(i)p Fr(g)p Fx(\))408 b(also)i(that)e(with)h Fv(k)453 b Fx(v)-36 b(ertices,)415 b(eac)-36 b(h)408 b(with)h(a)f(lo)36 b(op)410 b(attac)-36 b(hed,)413 b(and)-2350 21868 y(no)361 b(other)f(edges.)554 b(In)360 b(other)g(w)-36 b(ords,)375 b(in)360 b(the)g(sym)-36 b(b)36 b(ol)361 b(\(set)f(of)h(\\neigh)-36 b(b)36 b(ours")360 b(of)h(0\))f(for)h(a)g(circulan)-36 b(t)360 b(graph)-2350 23473 y(with)434 b(lo)36 b(ops,)435 b(also)f(0)g(ma)-36 b(y)434 b(app)36 b(ear.)-2350 25742 y Fu(Remark)486 b(8)1301 b Fx(The)422 b(\\basic")h(graphs)f(do)g(not)f(w)-36 b(ork)423 b(as)g(they)f(are;)k(the)c(request)g(for)g(the)g(relation)g(to)h(b)36 b(e)-2350 27347 y(transitiv)-36 b(e)537 b(forces)f(an)-36 b(y)536 b Fv(k)45 b Fx(-gon)536 b(to)g(\\b)36 b(ecome")537 b(a)f(complete)g(directed)f(graph)g(\(that)h(is,)562 b Fv(K)43314 27546 y Fs(k)44418 27347 y Fx(where)536 b(all)-2350 28952 y(edges)439 b(are)g(bidirected\):)588 b(b)-36 b(y)439 b(transitivit)-36 b(y)-108 b(,)441 b(connect)d(v)-36 b(ertices)440 b(at)f(distance)f(2,)j(then)d(at)h(distance)g(3)g(and) -2350 30557 y(so)391 b(on.)564 b(The)390 b(sup)36 b(erp)g(osition)390 b(of)h Fv(d)f Fx(copies)g(of)h Fv(K)20966 30764 y Fs(k)24 b(=d)22879 30557 y Fx(and)390 b Fv(l)420 b Fx(copies)390 b(of)h Fv(K)32510 30764 y Fs(k)24 b(=l)34231 30557 y Fx(b)36 b(ecomes)390 b(b)-36 b(y)390 b(transitivit)-36 b(y)391 b(the)-2350 32162 y(sup)36 b(erp)g(osition)433 b(of)g(GCD\()p Fv(d;)221 b(l)29 b Fx(\))434 b(copies)f(of)h Fv(K)19662 32369 y Ft(lcm\()p Fs(k)24 b(=d;k)g(=l)11 b Ft(\))24917 32162 y Fx(.)578 b(So)433 b(the)f(lattice)i(of)f (divisors)h(of)f Fv(k)478 b Fx(describ)36 b(es)433 b(all)-2350 33767 y(the)g(p)36 b(ossible)434 b(circulan)-36 b(t)434 b(transitiv)-36 b(e)434 b(digraphs,)f(that)g(is)h(top)36 b(ologies.)-399 35372 y(In)306 b(other)e(w)-36 b(ords,)332 b(a)305 b(top)36 b(ology)307 b(is)f(the)e(transitiv)-36 b(e)306 b(closure)g(of)g(a)f(union)g(of)h(cyclic)g(graphs;)348 b(its)306 b(incidence)-2350 36978 y(matrix)461 b(can)f(b)36 b(e)459 b(seen)h(as)g(the)f Fv(k)45 b Fx(th)459 b(p)36 b(o)-36 b(w)g(er)460 b(of)g(the)g(incidence)f(matrix)h(of)h(the)e (starting)h(graph)f(with,)467 b(as)-2350 38583 y(its)434 b(en)-36 b(tries,)434 b(b)36 b(o)g(olean)434 b(v)-72 b(ariables)434 b(0)g(and)f(1)h(\(so)g(that)f(1)296 b(+)e(1)370 b(=)e(1\).)-2350 40852 y Fu(Remark)385 b(9)1301 b Fx(By)334 b(acyclicit)-36 b(y)-108 b(,)356 b(for)335 b(eac)-36 b(h)334 b Fv(n)h Fx(the)f(only)h(circulan)-36 b(t)334 b(p)36 b(oset)334 b(is)g(the)g(one)g(with)h Fv(n)f Fx(incomparable) -2350 42457 y(elemen)-36 b(ts.)-2350 44726 y Fu(Remark)499 b(10)1301 b Fx(The)434 b(only)g(p)36 b(ossible)434 b(distinguished)e (sets)i(are)g(the)f(empt)-36 b(y)433 b(and)g(the)g(full)h(ones.)18336 46996 y(One)f(Last)g(Example)p Black Black 6182 49512 34937 45 v 6160 50840 45 1329 v 6847 50442 a Fl(Group)p 17064 50840 V 7328 w(P)-35 b(ark)g(er)425 b(sequence)p 27443 50840 V 1329 w(EIS)g(en)-35 b(try)p 34169 50840 V 1329 w(Notes)p 41096 50840 V 6182 50885 34937 45 v 6160 52213 45 1329 v 17064 52213 V 27443 52213 V 34169 52213 V 41096 52213 V 6160 53541 V 6847 53143 a Fk(S)7590 52741 y Fh(2)8455 53143 y Fj(\(pro)31 b(duct)369 b(action\))p 17064 53541 V 1331 w Fk(d)p Fj(\()p Fk(k)19367 52741 y Fh(2)19864 53143 y Fj(\))p 27443 53541 V 7835 w(A048691)p 34169 53541 V 2578 w(See)g(rem.)g(11)p 41096 53541 V 6160 54870 V 17064 54870 V 27443 54870 V 34169 54870 V 41096 54870 V 6182 54914 34937 45 v -2350 58181 a Fu(Remark)502 b(11)1301 b Fx(The)437 b(result)f(follo)-36 b(ws)439 b(from)e(Section)p 0 1 0 0 TeXcolorcmyk 436 w(2)p (#section.2) [[291 193 297 205] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)h(k)-36 b(eeping)437 b(in)f(mind)g(that)g Fv(d)p Fx(\()p Fv(k)40037 57699 y Ft(2)40563 58181 y Fx(\))g(is)h(equal)g(to)g (the)-2350 59786 y(n)-36 b(um)g(b)36 b(er)432 b(of)j(pairs)e(\()p Fv(i;)221 b(j)75 b Fx(\))434 b(suc)-36 b(h)432 b(that)i(lcm\()p Fv(i;)221 b(j)75 b Fx(\))369 b(=)f Fv(k)45 b Fx(.)-2350 64223 y Fy(Ac)-60 b(kno)g(wledgemen)g(ts)-2350 67143 y Fx(W)-108 b(e)437 b(thank)g(Dina)g(Ghinelli)f(and)g(P)-36 b(eter)437 b(J.)g(Cameron)g(for)g(help)f(and)g(encouragemen)-36 b(t.)587 b(W)-108 b(e)437 b(also)h(thank)-2350 68748 y(the)433 b(referee)h(for)h(useful)e(commen)-36 b(ts.)p Black Black eop %%Page: 10 10 10 9 bop Black 0 TeXcolorgray Black 40444 -272 a Fx(10)p Black -2350 3824 a Fy(References)p Black -164 6745 a Fx([1])p Black 651 w(P)-36 b(eter)512 b(J.)i(Cameron,)533 b(T)-108 b(ransitivit)-36 b(y)514 b(of)f(p)36 b(erm)-36 b(utation)512 b(groups)h(on)g(unordered)e(sets,)533 b Fw(Math.)k(Z.)1859 8350 y Fu(148)434 b Fx(\(1976\),)h(127{139.)p Black -164 11062 a([2])p Black 651 w(P)-36 b(eter)440 b(J.)h(Cameron,)h(Cohomological)i(asp)36 b(ects)440 b(of)h(t)-36 b(w)g(o-graphs,)442 b Fw(Math.)471 b(Z.)440 b Fu(157)h Fx(\(1977\),)i(101{)1859 12667 y(119.)p Black -164 15379 a([3])p Black 651 w(P)-36 b(eter)487 b(J.)h(Cameron,)502 b(P)-36 b(ortrait)488 b(of)g(a)g(t)-36 b(ypical)488 b(sum-free)g(set,) 501 b Fw(Surveys)514 b(in)g(Combinatorics)486 b Fx(\(C.)1859 16984 y(Whitehead,)323 b(ed.\),)h(13{42,)h(LMS)295 b(Lecture)g(Notes)h Fu(123)p Fx(,)325 b(Cam)-36 b(bridge)296 b(Univ.)g(Press,)324 b(Cam)-36 b(bridge,)1859 18589 y(1987.)p Black -164 21301 a([4])p Black 651 w(P)g(eter)393 b(J.)i(Cameron,)402 b(Some)394 b(treelik)-36 b(e)394 b(ob)72 b(jects,)403 b Fw(Quart.)428 b(J.)g(Math.)g(Oxfor)-66 b(d)428 b(Ser.)f(\(2\))395 b Fu(38)f Fx(\(1987\),)1859 22906 y(155{183.)p Black -164 25618 a([5])p Black 651 w(P)-36 b(eter)305 b(J.)g(Cameron,)332 b Fw(Oligomorphic)346 b(Permutation)g(Gr)-66 b(oups)p Fx(,)332 b(LMS)304 b(Lecture)g(Notes)i Fu(152)p Fx(,)331 b(Cam-)1859 27223 y(bridge)433 b(Univ.)h(Press,)g(Cam)-36 b(bridge,)434 b(1990.)p Black -164 29935 a([6])p Black 651 w(P)-36 b(eter)333 b(J.)h(Cameron,)354 b(Sequences)333 b(realized)h(b)-36 b(y)333 b(oligomorphic)i(p)36 b(erm)-36 b(utation)333 b(groups,)353 b Fw(J.)373 b(Inte)-66 b(ger)1859 31541 y(Se)g(q.)432 b Fu(3)h Fx(\(2000\),)i(00.1.5)h([)p 0 1 0 0 TeXcolorcmyk Fa (http://www.math.uwaterloo.ca/JIS/VOL3/groups.html)p [[200 433 502 445] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterloo.ca/JIS/VOL3/groups.html) pdfm Black Fx(].)p Black -164 34253 a([7])p Black 651 w(Daniele)878 b(A.)g(Gewurz,)989 b(P)-36 b(ark)g(er)877 b(v)-36 b(ectors)878 b(and)f(cycle)i(indices)e(of)i(p)36 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