Partial permutohedra

Behrend, Roger E. ORCID: https://orcid.org/0000-0002-6143-7439, Castillo, Federico, Chavez, Anastasia, Diaz-Lopez, Alexander, Escobar, Laura, Harris, Pamela E. and Insko, Erik 2023. Partial permutohedra. [Online]. arXiv: arXiv. Available at: https://arxiv.org/abs/2207.14253

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Abstract

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers m and n, the partial permutohedron P(m,n) is the convex hull of all vectors in {0,1,...,n}^m whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of P(m,n), and our methods and results include the following. For any m and n, we obtain a bijection between the nonempty faces of P(m,n) and certain chains of subsets of {1,...,m}, thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the h-polynomial of P(m,n). For any m and n with n ≥ m-1, we use a pyramidal subdivision of P(m,n) to establish a recursive formula for the normalized volume of P(m,n), from which we then obtain closed expressions for this volume. We also use a sculpting process (in which P(m,n) is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of P(m,n) with arbitrary m and fixed n ≤ 3, the volume of P(m,4) with arbitrary m, and the Ehrhart polynomial of P(m,n) with fixed m ≤ 4 and arbitrary n ≥ m-1.

Item Type:	Website Content
Date Type:	Published Online
Status:	Submitted
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	arXiv
Funders:	Leverhulme Trust
Last Modified:	05 Dec 2023 15:39
URI:	https://orca.cardiff.ac.uk/id/eprint/162473

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