Ehrhart polynomials of partial permutohedra

Behrend, Roger E. ORCID: https://orcid.org/0000-0002-6143-7439 2024. Ehrhart polynomials of partial permutohedra. [Online]. arXiv. Available at: https://arxiv.org/abs/2403.06975

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Abstract

For positive integers m and n, the partial permutohedron P(m,n) is a certain integral polytope in R^m, which can be defined as the convex hull of the vectors from {0,1,...,n}^m whose nonzero entries are distinct. For n = m-1, P(m,m-1) is (after translation by (1,...,1)) the polytope P_m of parking functions of length m, and for n ≥ m, P(m,n) is combinatorially equivalent to an m-stellohedron. The main result of this paper is an explicit expression for the Ehrhart polynomial of P(m,n) for any m and n with n ≥ m-1. The result confirms the validity of a conjecture for this Ehrhart polynomial in arXiv:2207.14253, and the n = m-1 case also answers a question of Stanley regarding the number of integer points in P_m. The proof of the result involves transforming P(m,n) to a unimodularly equivalent polytope in R^{m+1}, obtaining a decomposition of this lifted version of P(m,n) with n ≥ m-1 as a Minkowski sum of dilated coordinate simplices, applying a result of Postnikov for the number of integer points in generalized permutohedra of this form, observing that this gives an expression for the Ehrhart polynomial of P(m,n) with n ≥ m-1 as an edge-weighted sum over graphs (with loops and multiple edges permitted) on m labelled vertices in which each connected component contains at most one cycle, and then applying standard techniques for the enumeration of such graphs.

Item Type:	Website Content
Date Type:	Published Online
Status:	Unpublished
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	arXiv
Date of Acceptance:	2024
Last Modified:	22 Mar 2024 12:27
URI:	https://orca.cardiff.ac.uk/id/eprint/167322

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