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arXiv · 1606.03007

Euler-Mahonian statistics and descent bases for semigroup algebras

Benjamin Braun·McCabe Olsen
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Abstract

We consider quotients of the unit cube semigroup algebra by particular $\mathbb{Z}_r\wr S_n$-invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations $(π,ε)\in\mathbb{Z}_r\wr S_n$ and each element encodes the negative descent and negative major index statistics on $(π,ε)$. 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.

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Benjamin Braun, McCabe Olsen. 2017-10-15. Euler-Mahonian statistics and descent bases for semigroup algebras. https://arxiv.org/abs/1606.03007

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