\documentclass[12pt]{article}
\usepackage{amsmath,mathrsfs}
\usepackage{amssymb}
\textwidth=4.825in
\overfullrule=0pt
\thispagestyle{empty}

\begin{document}
\noindent
%
%
{\bf Edward A. Bender and E. Rodney Canfield}
%
%

\medskip
\noindent
%
%
{\bf Locally Restricted Compositions III. Adjacent-Part Periodic Inequalities}
%
%

\vskip 5mm
\noindent
%
%
%
%
We study compositions $c_1,\dots,c_k$
of the integer $n$ in which adjacent parts may be constrained to
satisfy some periodic inequalities, for example
$$
c_{2i}>c_{2i+1}<c_{2i+2}
~~\mbox{(alternating compositions).}
$$
The types of inequalities considered are $<$, $\le$, $>$, $\ge$
and $\ne$.
We show how to obtain generating functions from which various
pieces of asymptotic information can be computed.
There are asymptotically $Ar^{-n}$ compositions of $n$.
In a random uniformly selected composition of $n$, the
largest part and number of distinct parts are almost surely
asymptotic to $\log_{1/r}(n)$.
The length of the longest run is almost surely asymptotic
to $C\log_{1/r}(n)$ where C is an easily determined
rational number.
Many other counts are asymptotically normally distributed.
We present some numerical results for the various types of
alternating compositions.

\end{document}