Elena Tielker | Searcharxiv

arXiv2025

Weighted Ehrhart series and a type-$\mathsf{B}$ analogue of a formula of MacMahon

We present a formula for a generalisation of the Eulerian polynomial, namely the generating polynomial of the joint distribution of major index and descent statistic over the set of signed multiset permutations. It has a description in terms of the $h^*$-polynomial of a certain polytope. Moreover, we associate a family of polytopes to (generalised) Eulerian polynomials of types $\mathsf{A}$ and $\mathsf{B}$. Using this connection, properties of the generalised Eulerian numbers of types $\mathsf{A}$ and $\mathsf{B}$, such as palindromicity and unimodality, are reflected in certain properties of the associated polytope. We also present results on generalising the connection between descent polynomials and polytopes to coloured (multiset) permutations.

math.CO↗

arXiv2021

On Denert's statistic

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.

math.CO↗