Equivalence of the Descents Statistic on Some (4,4)-Avoidance Classes of Permutations

TL;DR

This paper establishes the equivalence of certain permutation and inversion sequence statistics within specific avoidance classes, providing new combinatorial insights and a bivariate refinement of large Schr"oder numbers.

Contribution

It demonstrates the equivalence of joint distributions of permutation and inversion sequence statistics in avoidance classes, confirming a recent conjecture and introducing a new bivariate enumeration.

Findings

Distribution equivalence between permutations and inversion sequences

Enumeration of classes by large Schr"oder numbers

New bivariate refinement of Schr"oder numbers

Abstract

In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the $n$-th large Schr\"oder number and thus one obtains a new bivariate refinement of these numbers as a consequence. We make use of auxiliary combinatorial statistics, special generating functions (specific to each class) and the kernel method to establish our results. In some cases, we utilize the conjecture itself in a creative way to aid in solving the system of functional equations satisfied by the associated generating functions.