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\vskip 1cm{\LARGE\bf Counting Dyck Paths According to the \\
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Maximum Distance Between Peaks and \\
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Valleys} \vskip 1cm
\large Toufik Mansour\\
Department of Mathematics\\
University of Haifa \\
31905 Haifa  \\
Israel\\
\href{mailto:toufik@math.haifa.ac.il}{\tt toufik@math.haifa.ac.il}\\
\ \\
Mark Shattuck\\
Department of Mathematics\\
University of Tennessee \\
Knoxville, TN 37996\\
USA \\
\href{mailto:shattuck@math.utk.edu}{\tt shattuck@math.utk.edu}\\
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\def\DD{\mathcal{D}}
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\begin{abstract}
A \emph{Dyck path} of length $2n$ is a lattice path from $(0,0)$ to
$(2n,0)$ consisting of up-steps $u=(1,1)$ and down-steps $d=(1,-1)$
which never passes below the $x$-axis.  Let $\DD_n$ denote the set of
Dyck paths of length $2n$.  A \emph{peak} is an occurrence of $ud$ (an
upstep immediately followed by a downstep) within a Dyck path, while a
\emph{valley} is an occurrence of $du$.  Here, we compute explicit
formulas for the generating functions which count the members of
$\DD_n$ according to the maximum number of steps between any two peaks,
any two valleys, or a peak and a valley. In addition, we provide closed
expressions for the total value of the corresponding statistics taken
over all of the members of $\DD_n$.  Equivalent statistics on the set
of $231$-avoiding permutations of length $n$ are also described.
\end{abstract}

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