## Abstract

Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular triangulation of the hypersimplex restricts to a triangulation of each r-stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an n-set, we provide a shelling of this triangulation that sequentially shells each r-stable sub-hypersimplex. In this case, we utilize the shelling to compute the Ehrhart h^{⁎}-polynomials of these polytopes, and the hypersimplex, via independence polynomials of graphs. For one such r-stable hypersimplex, this computation yields a connection to CR mappings of Lens spaces via Ehrhart–MacDonald reciprocity.

Original language | English |
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Pages (from-to) | 349-388 |

Number of pages | 40 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 157 |

DOIs | |

State | Published - Jul 2018 |

### Bibliographical note

Funding Information:Benjamin Braun was partially supported by the National Security Agency through awards H98230-13-1-0240 and H98230-16-1-0045. Liam Solus was partially supported by a 2014 National Science Foundation/Japan Society for the Promotion of Science East Asia and Pacific Summer Institute Fellowship (EAPSI award 1414621).

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- Ehrhart h-vector
- Hypersimplex
- Shelling
- Triangulation
- Unimodal
- r-stable hypersimplex

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics