Permutation patterns and statistics

Let SnSn denote the symmetric group of all permutations of {1,2,…,n}{1,2,…,n} and let S=∪n≥0SnS=∪n≥0Sn. If Π⊆SΠ⊆S is a set of permutations, then we let Avn(Π) be the set of permutations in SnSn which avoid every permutation of ΠΠ in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where ΠΠ and Π′Π′ are Wilf equivalent   if #Avn(Π)=#Avn(Π′) for all n≥0n≥0. In a recent paper, Sagan and Savage proposed studying a qq-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′Π,Π′st-Wilf equivalent   if Fnst(Π;q)=Fnst(Π′;q) for all n≥0n≥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⊆S3Π⊆S3. This leads us to consider various qq-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.