# S-hypersimplices, pulling triangulations, and monotone paths

## TL;DR

This paper explores the structure and dissections of S-hypersimplices, revealing their relation to multipermutahedra and demonstrating properties of triangulations similar to cubes.

## Contribution

It introduces new insights into faces, dissections, and monotone path polytopes of S-hypersimplices, connecting them to multipermutahedra and analyzing triangulation properties.

## Key findings

- Monotone path polytopes of S-hypersimplices produce all multipermutahedra types.
- Number of simplices in a pulling triangulation of a halfcube is order-independent.
- Faces and dissections of S-hypersimplices are characterized and related to classical polytopes.

## Abstract

An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.