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{\bf Edward A. Bender and E. Rodney Canfield}
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{\bf Locally Restricted Compositions II. General Restrictions and Infinite Matrices}
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We study compositions $\vec{\bf c}=(c_1,\dots,c_k)$ of the integer $n$ in
which the value $c_i$ of the $i$th part is constrained based on
previous parts within a fixed distance of $c_i$.  The constraints may
depend on $i$ modulo some fixed integer $m$.  Periodic constraints
arise naturally when $m$-rowed compositions are written in a single
row.  We show that the number of compositions of $n$ is asymptotic to
$Ar^{-n}$ for some $A$ and $r$ and that many counts can be expected to
have a joint normal distribution with means vector and covariance
matrix asymptotically proportional to $n$.  Our method of proof relies
on infinite matrices and does not readily lead to methods for accurate
estimation of the various parameters.  We obtain information about the
longest run.  In many cases, we obtain almost sure asymptotic
estimates for the maximum part and number of distinct parts.



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