Major index distribution over permutation classes

TL;DR

This paper investigates the distribution of the major index over pattern-avoiding permutations, showing polynomial growth in the number of such permutations with fixed major index as permutation length increases.

Contribution

It establishes the monotonicity of the distribution for singleton pattern sets and characterizes the asymptotic polynomial behavior of permutation counts.

Findings

For singleton pattern sets, the distribution values are monotonic with permutation length.

The number of pattern-avoiding permutations with fixed major index grows as a polynomial in n.

Degrees of these polynomials are determined for many pattern sets.

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 $M_n^m(\Pi) \leq M_{n+1}^m(\Pi)$. 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.