Counting pattern-avoiding permutations by big descents

TL;DR

This paper investigates the distribution of big descents in permutations avoiding certain length-three patterns, classifies pattern sets into equivalence classes, and derives formulas using generating functions and combinatorial bijections.

Contribution

It classifies pattern sets into bdes-Wilf equivalence classes and provides formulas for big descent distributions using combinatorial and generating function methods.

Findings

Classified all size-1 and size-2 pattern sets into bdes-Wilf equivalence classes.

Derived explicit formulas for big descent distributions for each class.

Proposed future research directions including conjectures on real-rootedness and log-concavity.

Abstract

A descent $k$ of a permutation $\pi=\pi_{1}\pi_{2}\dots\pi_{n}$ is called a big descent if $\pi_{k}>\pi_{k+1}+1$; denote the number of big descents of $\pi$ by $\operatorname{bdes}(\pi)$. We study the distribution of the $\operatorname{bdes}$ statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we classify all pattern sets $\Pi\subseteq\mathfrak{S}_{3}$ of size 1 and 2 into $\operatorname{bdes}$-Wilf equivalence classes, and we derive a formula for the distribution of big descents for each of these classes. Our methods include generating function techniques along with various bijections involving objects such as Dyck paths and binary words. Several future directions of research are proposed, including conjectures concerning real-rootedness, log-concavity, and Schur positivity.