Séminaire Lotharingien de Combinatoire, 89B.64 (2023), 12 pp.
Roger E. Behrend, Federico Castillo, Anastasia Chavez, AAlexnder
Diaz-Lopez, Laura Escobar, Pamela Harris and Erik Insko]
Partial Permutohedra
Abstract.
Partial permutohedra are lattice polytopes which were recently introduced
and studied by Heuer and Striker [arXiv:2012.09901].
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.
We 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.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
The following versions are available: