Summary: Let \(\mathcal S_{n}\) denote the symmetric group of all permutations of \({1,2,\ldots ,n}\) and let \(\mathcal S=\cup _{n\geq 0}\mathcal S_{n}\). If \(\varPi \subseteq \mathcal S\) is a set of permutations, then we let \(Av_{n}(\varPi)\) be the set of permutations in \(\mathcal S_{n}\) which avoid every permutation of \(\varPi \) in the sense of pattern avoidance.
One of the celebrated notions in pattern theory is that of Wilf-equivalence, where \(\varPi \) and \(\varPi^{\prime}\) are Wilf equivalent if \(\#Av_{n}(\varPi)=\#Av_{n}(\varPi^{\prime})\) for all \(n\geq 0\).
In a recent paper, B. E. Sagan and C. D. Savage [“Mahonian pairs”, J. Comb. Theory, Ser. A 119, No.3, 526–545 (2012; Zbl 1242.05019)] proposed studying a \(q\)-analogue of this concept defined as follows. Suppose \(\text{st}:\mathcal S \to \{0,1,2,\ldots\}\) is a permutation statistic and consider the corresponding generating function \(F_n^{\text{st}} (\varPi;q)=\sum_{\sigma\in Av_n(\varPi)}q^{\text{st}\sigma}\). Call \(\varPi ,\varPi^{\prime}\)st-Wilf equivalent if \(F_n^{\text{st}} (\varPi ;q)=F_n^{\text{st}} (\varPi';q)\) for all \(n\geq 0\).
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 \(\varPi \subseteq \mathcal S_{3}\). 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 second fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article.