Permutation Statistics and Pattern Avoidance in Involutions

Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern $\pi\in\mathfrak{S}_k$ if $\sigma$ has a subsequence of length $k$ whose letters are in the same relative order as $\pi$. This paper is a comprehensive study of the same two statistics, number of inversions and major index, over involutions $\mathcal{I}_n=\{\sigma\in\mathfrak{S}_n:\sigma^2=\text{id}\}$ that avoid one or more length three patterns. The equalities between the generating functions are consequently determined via symmetries and we conjecture this happens for longer patterns as well. We describe the generating functions for each set of patterns including the fixed-point-free case, $\sigma(i)\neq i$ for all $i.$ Notating $M\mathcal{I}_n(\pi)$ as the generating function for the major index over the avoidance class of involutions associated to $\pi$ we particularly present an independent determination that $M\mathcal{I}_n(321)$ is the $q$-analogue for the central binomial coefficient that first appeared in a paper by Barnebei, Bonetti, Elizalde and Silimbani. A shorter proof is presented that establishes a connection to core, a central topic in poset theory. We also prove that $M\mathcal{I}_n(132;q)=q^{\binom{n}{2}}M\mathcal{I}_n(213;q^{-1})$ and that the same symmetry holds for the larger class of permutations conjecturing that the same equality is true for involutions and permutations given any pair of patterns of the form $k(k-1)\dots 1(k+1)(k+2)\dots m$ and $12\dots (k-1) m(m-1)\dots k$, $k\leq m$.