## Abstract

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) h^{⁎}-polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal h^{⁎}-polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Mustaţǎ and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are h^{⁎}-unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.

Original language | English |
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Pages (from-to) | 122-142 |

Number of pages | 21 |

Journal | Advances in Applied Mathematics |

Volume | 100 |

DOIs | |

State | Published - Sep 2018 |

### Bibliographical note

Funding Information:Liam Solus was partially supported by an NSF Division of Mathematical Sciences Postdoctoral Fellowship (DMS – 1606407).

Publisher Copyright:

© 2018 Elsevier Inc.

## ASJC Scopus subject areas

- Applied Mathematics