Restricted permutations refined by number of crossings and nestings

TL;DR

This paper classifies permutations avoiding certain patterns based on crossings and nestings, introduces a recursive bijection preserving these statistics, and explores implications for q,p-Catalan numbers.

Contribution

It provides a direct recursive definition of a bijection between 321- and 132-avoiding permutations that preserves crossings and nestings statistics.

Findings

The bijection preserves the number of crossings.

The bijection preserves the number of fixed points and excedances.

Results relate to q,p-Catalan numbers and their properties.

Abstract

Let $st=\{st_1,\ldots,st_k\}$ be a set of $k$ statistics on permutations with $k\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in $st$ over the sets of $T$-avoiding permutations $S_n(T)$ and $T'$-avoiding permutations $S_n(T')$ are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in $S_3$, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection $\Theta:S_n(321)\rightarrow S_n(132)$ which was originally exhibited by Elizalde and Pak in \cite{ElizP}. They proved that the bijection $\Theta$ preserves the number of fixed points and excedances. Since the given formulation of $\Theta$ is not direct, we show that it can be defined directly by a recursive formula. Then, we prove that it also preserves the number of crossings. Due to the fact that the sets of non-nesting permutations and 321-avoiding permutations are the same, these properties of the bijection $\Theta$ leads to an unexpected result related to the q,p-Catalan numbers of Randrianarivony defined in \cite{ARandr}.