Mahonian statistics on words with fixed weak right-to-left minima and on permutations with a fixed descent set

TL;DR

This paper proves equidistribution of various Mahonian and Euler--Mahonian statistics on words and permutations with fixed properties, and applies these results to related combinatorial structures.

Contribution

It establishes new equidistribution results for Mahonian statistics on words and permutations with fixed parameters, extending to related combinatorial objects.

Findings

Statistics within each class are equidistributed for words with fixed weak right-to-left minima.

Statistics within each class are equidistributed for permutations with fixed descent set.

Results apply to set partitions, 221-avoiding words, alternating permutations, and permutations with k alternating runs.

Abstract

Our first main result shows that, for words with a fixed multiset of weak right-to-left minima, the statistics within each of the following three classes are equidistributed: 1. Mahonian statistics: $\textsf{inv}$, $\textsf{maj}$, $\textsf{den}$, $\textsf{mak}$, $\textsf{mad}$, $\textsf{inv}_{r}$, $r\textsf{maj}$, and $r\textsf{den}$; 2. Euler--Mahonian statistics: $(\textsf{des},\textsf{maj})$, $(\textsf{exc},\textsf{den})$, and $(\textsf{des},\textsf{mak})$; 3. $r$-Euler--Mahonian statistics: $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$. Our second main result shows that, for permutations with a fixed descent set, the statistics within each of the following two classes are equidistributed: 1. Mahonian statistics: $\textsf{inv}$, $\textsf{imaj}$, $\textsf{imak}$, $\textsf{iinv}_{r}$, and $\textsf{istat}$; 2. Mahonian--Stirling-type statistics: $(\textsf{inv},\textsf{lrmax})$, $(\textsf{imaj},\textsf{lrmax})$, $(\textsf{imak},\textsf{lrmax})$, and $(\textsf{iinv}_{r},\textsf{lrmax})$. Moreover, we apply these results to set partitions, 221-avoiding words, alternating permutations, and permutations with $k$ alternating runs, thereby obtaining several families of equidistributed Mahonian-type statistics on these combinatorial structures.