Joint distributions of statistics over permutations avoiding two patterns of length 3

TL;DR

This paper derives explicit formulas for joint distributions of multiple permutation statistics over classes avoiding two patterns of length 3, extending previous work and providing rational generating functions and combinatorial proofs.

Contribution

It generalizes existing results by providing explicit formulas for joint distributions of six and four permutation statistics over pattern-avoiding classes, including new extensions to stack-sortable permutations.

Findings

Explicit formulas for joint distributions of six permutation statistics.

Explicit formulas for joint distributions of four permutation statistics.

All generating functions are rational and combinatorial proofs are provided.

Abstract

Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any two patterns of length 3. In this paper, we generalize these results in two different ways: we find explicit formulas for the joint distribution of six statistics (asc, des, lrmax, lrmin, rlmax, rlmin), and also explicit formulas for the joint distribution of four statistics (asc, des, MNA, MND) on these permutations in all cases. The latter result also extends the recent studies by Kitaev and Zhang of the statistics MNA and MND (related to non-overlapping occurrences of ascents and descents) on stack-sortable permutations. All multivariate generating functions in our paper are rational, and we provide combinatorial proofs of five equidistribution results that can be derived from the generating functions.