Restricted permutations refined by number of crossings and nestings

Abstract

Let st={st1,…,stk} be a set of k statistics on permutations with k≥1. We say that two given subsets 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 Sn(T) and T′-avoiding permutations Sn(T′) are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in S3, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection Θ:Sn(321)→Sn(132) which was originally exhibited by Elizalde and Pak in Elizalde and Pak (2004). They proved that the bijection Θ preserves the number of fixed points and excedances. Since the given formulation of Θ 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 Θ lead to an unexpected result related to the q,p-Catalan numbers of Randrianarivony defined in Randrianarivony (1998).