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

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: $\mathsf{\text{inv}}$, $\mathsf{\text{maj}}$, $\mathsf{\text{den}}$, $\mathsf{\text{mak}}$, $\mathsf{\text{mad}}$, ${\mathsf{\text{inv}}}_r$, $r\mathsf{\text{maj}}$, and $r\mathsf{\text{den}}$; 2. Euler–Mahonian statistics: $(\mathsf{\text{des}},\mathsf{\text{maj}})$, $(\mathsf{\text{exc}},\mathsf{\text{den}})$, and $(\mathsf{\text{des}},\mathsf{\text{mak}})$; 3. $r$-Euler–Mahonian statistics: $(r\mathsf{\text{des}},r\mathsf{\text{maj}})$ and $(r\mathsf{\text{exc}},r\mathsf{\text{den}})$. 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: $\mathsf{\text{inv}}$, $\mathsf{\text{imaj}}$, $\mathsf{\text{imak}}$, ${\mathsf{\text{iinv}}}_r$, and $\mathsf{\text{istat}}$; 2. Mahonian–Stirling-type statistics: $(\mathsf{\text{inv}},\mathsf{\text{lrmax}})$, $(\mathsf{\text{imaj}},\mathsf{\text{lrmax}})$, $(\mathsf{\text{imak}},\mathsf{\text{lrmax}})$, and $({\mathsf{\text{iinv}}}_r,\mathsf{\text{lrmax}})$. 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.