Restricted permutations refined by number of crossings and nestings

Paul M. Rakotomamonjy

Published 2018-08-11Version 1

Let $st=\{st_1,st_2,...,st_k\}$ 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 pattern 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 $\Theta$ from the set of 321-avoiding permutations $S_n(321)$ to the set of 132-avoiding permutations $S_n(132)$ which was exhibited by Elizalde and Pak. 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. These properties of the bijection $\Theta$ leads to an unexpected result related to the q,p-Catalan numbers defined by Randrianarivony.