Permutation patterns and statistics

TL;DR

This paper explores q-analogues of Wilf-equivalence in permutation pattern avoidance, focusing on inv and maj statistics, and classifies equivalences for permutations in S_3, connecting to various q-analogues of classical sequences.

Contribution

It provides the first detailed analysis of st-Wilf equivalences for inv and maj statistics, including classifications for permutations in S_3 and connections to q-analogues of classical sequences.

Findings

Classified all inv- and maj-Wilf equivalences in S_3.

Connected permutation statistics to q-analogues of Catalan, Fibonacci, and other numbers.

Answered a question about Mahonian pairs from previous research.

Abstract

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegative integers. Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth study of this concept for the inv and maj statistics. In particular, we determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This leads us to consider various q-analogues of the Catalan numbers, Fibonacci numbers, triangular numbers, and powers of two. Our proof techniques use lattice paths, integer partitions, and Foata's fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article.