Euler-Mahonian statistics and descent bases for semigroup algebras

TL;DR

This paper introduces a new algebraic basis for certain semigroup algebra quotients, linking it to colored permutation statistics and enabling the derivation of combinatorial identities related to Euler-Mahonian distributions.

Contribution

It provides an algebraic interpretation of negative descent and major index statistics on colored permutations using Gr"obner basis methods.

Findings

Established a basis indexed by colored permutations encoding specific statistics.

Connected algebraic structures to combinatorial Euler-Mahonian distributions.

Recovered known combinatorial identities through algebraic methods.

Abstract

We consider quotients of the unit cube semigroup algebra by particular $\mathbb{Z}_r\wr S_n$-invariant ideals. Using Gr\"obner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations $(\pi,\epsilon)\in\mathbb{Z}_r\wr S_n$ and each element encodes the negative descent and negative major index statistics on $(\pi,\epsilon)$. This gives an algebraic interpretation of these statistics which was previously unknown. This basis of the $\mathbb{Z}_r\wr S_n$-quotients allows us to recover certain combinatorial identities involving Euler-Mahonian distributions of statistics.