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 2025. Partial Permutohedra. Discrete & Computational Geometry 10.1007/s00454-025-00755-0 Item availability restricted.

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Abstract

Partial permutohedra are lattice polytopes recently introduced and studied by Heuer and Striker. For positive integers m and n, the partial permutohedron P(m, n) is defined as the convex hull of all vectors in the set {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 establish a bijection between the nonempty faces of P(m, n) and certain chains of subsets of the set {1, ..., m}, thereby confirming a conjecture of Heuer and Striker. Using this characterization of faces, we derive a closed-form expression for the h-polynomial of P(m, n). For any m and n such that n ≥ m − 1, we apply a pyramidal subdivision of P(m, n) to establish a recursive formula for its normalized volume, from which we also obtain closed-form expressions for this volume. We introduce a sculpting process, where P(m, n) is constructed by successively removing specific regions from a simplex or hypercube, to derive: Closed-form expressions for the Ehrhart polynomial of P(m, n) for arbitrary m and fixed n ≤ 3, The normalized volume of P(m, 4) for arbitrary m, and The Ehrhart polynomial of P(m, n) for fixed m ≤ 4 and arbitrary n ≥ m − 1.

Item Type:	Article
Date Type:	Published Online
Status:	In Press
Schools:	Schools > Mathematics
Publisher:	Springer
ISSN:	0179-5376
Funders:	Leverhulme Trust
Date of First Compliant Deposit:	22 October 2025
Date of Acceptance:	13 June 2025
Last Modified:	23 Oct 2025 12:30
URI:	https://orca.cardiff.ac.uk/id/eprint/181833

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