Permutation patterns and statistics

Abstract

Let Sn denote the symmetric group of all permutations of {1,2,…,n} and let S=∪n≥0Sn. If Π⊆S is a set of permutations, then we let Avn(Π) be the set of permutations in Sn which avoid every permutation of Π in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Π and Π′ are Wilf equivalent if #Avn(Π)=#Avn(Π′) for all n≥0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S→{0,1,2,…} is a permutation statistic and consider the corresponding generating function Fnst(Π;q)=∑σ∈Avn(Π)qstσ. Call Π,Π′st-Wilf equivalent if Fnst(Π;q)=Fnst(Π′;q) for all n≥0. 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 Π⊆S3. 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 second fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan–Savage article.