On Denert's statistic

TL;DR

This paper links Denert's genus zeta function numerators to Euler-Mahonian statistics on multiset permutations, establishing reciprocity properties and identities involving Hadamard products, and introduces Mahonian companions for permutation groups.

Contribution

It provides a new combinatorial description of genus zeta function numerators and explores their algebraic and reciprocity properties, extending the understanding of local hereditary orders.

Findings

Numerators described via joint distribution of Euler-Mahonian statistics.

Reciprocity property for genus zeta functions of rectangular compositions.

Identity involving Hadamard products of genus zeta functions.

Abstract

We show that the numerators of genus zeta function associated with local hereditary orders studied by Denert can be described in terms of the joint distribution of Euler-Mahonian statistics on multiset permutations defined by Han. We use this result to deduce a reciprocity property for genus zeta functions of local hereditary orders whose associated composition is a rectangle. We also record a remarkable identity satisfied by genus zeta functions of local hereditary orders in terms of Hadamard products of genus zeta functions of maximal orders. Finally, we define Mahonian companions of excedance statistics on groups of signed and even-signed permutations.