Major index distribution over permutation classes

Abstract

For a permutation $\pi$ the major index of $\pi$ is the sum of all indices $i$ such that ${\pi }_i>{\pi }_{i+1}$. It is well known that the major index is equidistributed with the number of inversions over all permutations of length $n$. In this paper, we study the distribution of the major index over pattern-avoiding permutations of length $n$. We focus on the number $M_n^m(\Pi )$ of permutations of length $n$ with major index $m$ and avoiding the set of patterns $\Pi$. First we are able to show that for a singleton set $\Pi =\{\sigma \}$ other than some trivial cases, the values $M_n^m(\Pi )$ are monotonic in the sense that $M_n^m(\Pi )\le M_{n+1}^m(\Pi )$. Our main result is a study of the asymptotic behaviour of $M_n^m(\Pi )$ as $n$ goes to infinity. We prove that for every fixed $m$ and $\Pi$ and $n$ large enough, $M_n^m(\Pi )$ is equal to a polynomial in $n$ and moreover, we are able to determine the degrees of these polynomials for many sets of patterns.