Weighted Ehrhart series and a type-$\mathsf{B}$ analogue of a formula of MacMahon
Abstract
We present a formula for a generalisation of the Eulerian polynomial, namely the joint distribution of major index and descent statistic on signed multiset permutations. It allows a description in terms of the $h^*$-polynomial of a certain polytope. We associate a family of polytopes to (generalised) permutations of types $\mathsf{A}$ and $\mathsf{B}$. We use this connection to study properties of the (generalised) Eulerian numbers, such as palindromicity and unimodality, by identifying certain properties of the associated polytope. We also present partial results on generalising the connection between descent polynomials and polytopes to coloured (multiset) permutations.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.15774
- arXiv:
- arXiv:2203.15774
- Bibcode:
- 2022arXiv220315774T
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- An extended abstract of this paper has been accepted in FPSAC 2022