Equidistribution and Sign-Balance on 321-Avoiding Permutations

TL;DR

This paper investigates properties of 321-avoiding permutations, establishing equidistribution of certain statistics and relating sign-and-last-descent enumerators across different permutation sizes using recursive generating functions.

Contribution

It introduces new equidistribution results for last descent and index-minus-one statistics, and connects sign-and-last-descent enumerators across permutation sizes.

Findings

Last descent and last index minus one are equidistributed in 321-avoiding permutations.

Sign-and-last-descent enumerators for even and odd sizes relate to smaller permutations.

Results extend to Dyck paths, showing broader combinatorial connections.

Abstract

Let $T_n$ be the set of 321-avoiding permutations of order $n$. Two properties of $T_n$ are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over $T_n$, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for Dyck paths. (2) The sign-and-last-descent enumerators for $T_{2n}$ and $T_{2n+1}$ are essentially equal to the last-descent enumerator for $T_n$. The proofs use a recursion formula for an appropriate multivariate generating function.