Partial permutohedra
Roger E. Behrend, Federico Castillo, Anastasia Chavez, Alexander Diaz-Lopez, Laura Escobar, Pamela E. Harris, Erik Insko

TL;DR
This paper investigates the geometric and combinatorial properties of partial permutohedra, including face structure, volume, and Ehrhart polynomial, providing explicit formulas and confirming conjectures in the field.
Contribution
It establishes a face lattice bijection, confirms a conjecture, and derives closed-form expressions for volume and Ehrhart polynomial of partial permutohedra.
Findings
Bijection between faces and chains of subsets of {1,...,m}
Closed formulas for volume and Ehrhart polynomial
Confirmed conjecture of Heuer and Striker
Abstract
Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers and , the partial permutohedron is the convex hull of all vectors in whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of , and our methods and results include the following. For any and , we obtain a bijection between the nonempty faces of and certain chains of subsets of , thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the -polynomial of . For any and with , we use a pyramidal subdivision of to establish a recursive formula for the normalized volume of…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · graph theory and CDMA systems · Point processes and geometric inequalities
