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Math 491, Problem Set \#15 \\
(due 11/18/03)
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\begin{enumerate}
\item[(a)]
Let $A_n$ be the average number of times
that a $2n$-step Dyck path
returns to the origin
(counting $(2n,0)$ as a return but not $(0,0)$),
so that $A_0 = 0$, $A_1 = 1$, $A_2 = 3/2$, and $A_3 = 9/5$.
Use Maple to compute $A_n$
for various small values of $n$ (1 through 6, at least),
and conjecture a general formula.
\item[(b)]
Give an algebraic proof of your conjecture using generating functions.
\item[(c)]
Give a bijective proof of your conjecture,
using the relationship between
Dyck paths and triangulations of polygons.
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