# Lipschitz polytopes of posets and permutation statistics

## TL;DR

This paper explores Lipschitz functions on finite posets, characterizes the geometry of associated polytopes using permutation statistics, and introduces generalized hypersimplices with combinatorial volume interpretations.

## Contribution

It introduces Lipschitz polytopes for posets, links their geometry to permutation statistics, and defines generalized hypersimplices with new combinatorial insights.

## Key findings

- Lipschitz polytopes are centrally-symmetric and Gorenstein for ranked posets.
- Permutation statistics generalize descents and ascents within this framework.
- Volumes and $h^*$-vectors of generalized hypersimplices have combinatorial interpretations.

## Abstract

We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that generalize descents and big ascents. For ranked posets, Lipschitz polytopes are centrally-symmetric and Gorenstein, which implies symmetry and unimodality of the statistics. Finally, we define $(P,k)$-hypersimplices as generalizations of classical hypersimplices and give combinatorial interpretations of their volumes and $h^*$-vectors.