Further refinements of Euler-Mahonian statistics for multipermutations

Permutation statistics constitute a classical subject of enumerative combinatorics. In her study of the genus zeta function, Denert discovered a new Mahonian statistic for permutations, which is called the Denert’s statistic ({\bf $\den$}) by Foata and Zeilberger. As natural extensions of the $r$-descent number ({\bf $r\des$}) and the $r$-major index ({\bf $r\maj$}) introduced by Rawlings, Liu introduced the $g$-gap $\ell$-level descent number ({\bf $g\des_{\ell}$}) and the $g$-gap $\ell$-level major index ({\bf $g\maj_{\ell}$}) for permutations. In this paper, we introduce the $g$-gap $\ell$-level Denert’s statistic ({\bf $g\den_{\ell}$}) and the $g$-gap $\ell$-level excedance number ({\bf $g\exc_{\ell}$}) for multipermutations, which serve as natural generalizations of the Denert’s statistic ({\bf $\den$}) and the excedance number ({\bf $\exc$}) for multipermutations first introduced by Han. By constructing two explicit bijections, we establish the equidistribution of the pairs $(g\exc_{\ell}, g\den_{h} )$ and $(g\des_{\ell}, g\maj_{\ell})$ over multipermutations for all $1\leq h\leq g+\ell$. Our result provides a new proof of the equidistribution of the pairs ($\des$, $\maj$) and ($\exc$, $\den$) over multipermutations originally derived by Han and enables us to confirm a recent conjecture posed by Huang-Lin-Yan. Furthermore, we demonstrate that for all $1\leq h\leq g+\ell$, the pair $(g\exc_\ell, g\den_{h})$ is $r$-Euler-Mahonian over multipermutations of $M={1^k, 2^k, \ldots, n^k}$ where $r=g+\ell-1$ and $k\geq 1$, which extends a recent novel result derived by Liu from permutations to multipermutations.

💡 Research Summary

The paper develops a comprehensive theory of Euler‑Mahonian statistics for multipermutations, extending several classical permutation statistics to a more general “g‑gap ℓ‑level” framework. Starting from the well‑known descent set, descent number (des), major index (maj), and inversion number (inv), the authors recall MacMahon’s equidistribution of inv and maj, which holds for multipermutations as well. Building on Liu’s g‑gap ℓ‑level descent number g desℓ and major index g majℓ, they define analogous statistics for multipermutations without alteration.

The central novelty lies in introducing two new statistics: the g‑gap ℓ‑level Denert statistic g denℓ and the g‑gap ℓ‑level excedance number g excℓ. An index i is a g‑gap ℓ‑level excedance place if the entry α_i exceeds the i‑th entry x_i of the weakly increasing rearrangement of the word by at least g and also satisfies α_i ≥ x_i+ℓ. The statistic g excℓ simply counts such places. The statistic g denℓ is defined as the sum of the indices of g‑excedance places, plus the weak inversion count of the subsequence formed by the excedance letters, plus the ordinary inversion count of the subsequence formed by the non‑excedance letters. When g=ℓ=1 these reduce to the classical exc and Denert statistics.

To prove the main equidistribution results, the authors construct two explicit bijections: Φ_den^{g,h} and Φ_maj^{g,ℓ}. Both map a pair consisting of a multipermutation w (from a reduced multiset M₁) and a partition λ to a multipermutation in the original multiset M. The bijection Φ_den^{g,h} satisfies
 g den_h(Φ_den^{g,h}(w,λ)) = g den_h(w) + |λ|,
and it controls the change in g excℓ: if g excℓ(w)=s and g excℓ(Φ_den^{g,h}(w,λ))=t, then the partition size satisfies t−s = δ_ℓ + γ_g − 1, where δ_ℓ and γ_g are combinatorial constants depending on ℓ and g. Similarly, Φ_maj^{g,ℓ} preserves the relation
 g maj_ℓ(Φ_maj^{g,ℓ}(w,λ)) = g maj_ℓ(w) + |λ|,
and adjusts g desℓ in an analogous way.

Using these bijections, the authors first show that g den_h is Mahonian: its distribution over S_M coincides with that of inv. By induction on the number of distinct symbols, they prove that g den_h and inv have identical generating functions, which also implies the Mahonian property of g den_ℓ.

Combining the bijections with MacMahon’s theorem yields the fundamental equidistribution: for any positive integers g, ℓ and any h with 1 ≤ h ≤ g+ℓ,
 ∑{w∈S_M} t^{g des_ℓ(w)} q^{g maj_ℓ(w)} = ∑{w∈S_M} t^{g exc_ℓ(w)} q^{g den_h(w)}.
Consequently, the pair (g exc_ℓ, g den_h) is r‑Euler‑Mahonian over S_M, where r = g + ℓ – 1. This result simultaneously generalizes several known identities: setting g=ℓ=1 recovers Han’s (des, maj) ↔ (exc, den) equidistribution; choosing g=ℓ and h=ℓ reproduces Liu’s (g exc_ℓ, g den_h) equidistribution for ordinary permutations; and taking h=g+ℓ yields the conjecture of Huang‑Lin‑Yan concerning (r exc, r den).

The authors further extend the theory to the uniform multiset M = {1^k, 2^k, …, n^k} with arbitrary multiplicity k ≥ 1. By adapting the bijections to account for the repeated entries, they prove that the same r‑Euler‑Mahonian property holds for all k, thereby lifting Liu’s recent result from permutations to multipermutations.

The paper concludes with remarks on the algorithmic aspects of the bijections, noting that both Φ_den^{g,h} and Φ_maj^{g,ℓ} can be implemented recursively by inserting or removing parts of a partition, and suggesting future directions such as q‑analogs of the new statistics, refined generating functions, and connections to other combinatorial structures.

Overall, this work provides a unified and highly general framework for Euler‑Mahonian statistics on multipermutations, introduces natural g‑gap ℓ‑level extensions of Denert’s statistic and excedance, and resolves several open conjectures through constructive bijective proofs.