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

TL;DR

This paper generalizes Eulerian polynomials to signed multiset permutations, linking their properties to polytope geometry, and extends these connections to coloured permutations, revealing symmetry and unimodality features.

Contribution

It introduces a new formula for signed multiset permutation statistics and connects these to polytope $h^*$-polynomials, extending Eulerian polynomial theory to type-$ extsf{B}$ and coloured permutations.

Findings

Derived a formula for signed multiset permutation generating polynomials.

Connected Eulerian polynomials to polytope $h^*$-polynomials, revealing symmetry properties.

Extended descent polynomial-polytope relations to coloured permutations.

Abstract

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.