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{\bf Jason Bandlow and Kendra Killpatrick}
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{\bf An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations}
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The symmetric $q,t$-Catalan polynomial $C_n(q,t)$,
which specializes to the Catalan polynomial $C_n(q)$ when $t=1$, was
defined by Garsia and Haiman in 1994.  In 2000, Garsia and Haglund
described statistics $a(\pi)$ and $b(\pi)$ on Dyck paths such that
$C_n(q,t) = \sum_{\pi} q^{a(\pi)}t^{b(\pi)}$ where the sum is over all
$n \times n$ Dyck paths.  Specializing $t=1$ gives the Catalan
polynomial $C_n(q)$ defined by Carlitz and Riordan and further studied by Carlitz.  
Specializing both $t=1$ and $q=1$ gives the usual
Catalan number $C_n$.  The Catalan number $C_n$ is known to count the
number of $n \times n$ Dyck paths and the number of $312$-avoiding
permutations in $S_n$, as well as at least 64 other combinatorial
objects.  In this paper, we define a bijection between Dyck paths and
$312$-avoiding permutations which takes the area statistic $a(\pi)$ on
Dyck paths to the inversion statistic on $312$-avoiding permutations.
The inversion statistic can be thought of as the number of $(21)$
patterns in a permutation $\sigma$.  We give a characterization for
the number of $(321)$, $(4321)$, $\dots$, $(k\cdots21)$ patterns that
occur in $\sigma$ in terms of the corresponding Dyck path.


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