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{\bf Edward A. Bender and E. Rodney Canfield}
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{\bf Locally Restricted Compositions I. Restricted Adjacent Differences}
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We study compositions of the integer $n$ in which the first part,
successive differences, and the last part are constrained to lie in
prescribed sets ${\cal L,D,R}$, respectively.  A simple condition on
${\cal D}$ guarantees that the generating function $f(x,{\cal L,D,R})$
has only a simple pole on its circle of convergence and this at
$r({\cal D})$, a function independent of ${\cal L}$ and ${\cal R}$.
Thus the number of compositions is asymptotic to $Ar({\cal D})^{-n}$
for a suitable constant $A=A({\cal L,D,R})$.  We prove a multivariate
central and local limit theorem and apply it to various statistics of
random locally restricted compositions of $n$, such as number of
parts, numbers of parts of given sizes, and number of rises.  The
first and last parts are shown to have limiting distributions and to
be asymptotically independent.  If ${\cal D}$ has only finitely many
positive elements ${\cal D}^+$, or finitely many negative elements
${\cal D}^-$, then the largest part and number of distinct part sizes
are almost surely $\Theta((\log n)^{1/2})$.  On the other hand, when
both ${\cal D}^+$ and ${\cal D}^-$ have a positive asymptotic lower
``local log-density'', we prove that the largest part and number of
distinct part sizes are almost surely $\Theta(\log n)$, and we give
sufficient conditions for the largest part to be almost surely
asymptotic to $\log_{1/r({\cal D})}n$.

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