The equidistribution of some Mahonian statistics over permutations avoiding a pattern of length three

TL;DR

This paper proves that certain Mahonian and related permutation statistics are equally distributed over classes of permutations avoiding specific length-three patterns, confirming several conjectures from 2018.

Contribution

It establishes the equidistribution of multiple multistatistics over pattern-avoiding permutations, including Mahonian and generalized pattern-based statistics, solving several open conjectures.

Findings

Inv and foze' are equidistributed over certain pattern-avoiding classes.

Maj and makl are equidistributed over these classes.

Results confirm conjectures posed by Amini in 2018.

Abstract

We prove the equidistribution of several multistatistics over some classes of permutations avoiding a $3$-length pattern. We deduce the equidistribution, on the one hand of inv and foze" statistics, and on the other hand that of maj and makl statistics, over these classes of pattern avoiding permutations. Here inv and maj are the celebrated Mahonian statistics, foze" is one of the statistics defined in terms of generalized patterns in the 2000 pioneering paper of Babson and Steingr\'imsson, and makl is one of the statistics defined by Clarke, Steingr\'imsson and Zeng in 1997. These results solve several conjectures posed by Amini in 2018.