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

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.