Séminaire Lotharingien de Combinatoire, 91B.9 (2024), 12 pp.
Luis Ferroni and Benjamin Schröter
Enumerating The Faces of Split Matroid Polytopes
Abstract.
Computing f-vectors of polytopes is in general hard, and only little
is known about their shape. We initiate the study of properties of
f-vector of matroid base polytopes, by focusing on the class of
split matroids, i.e., matroid polytopes arising from compatible splits
of a hypersimplex. Unlike valuative invariants, the f-vector behaves
in a much more unpredictable way, and the modular pairs of cyclic
flats play a role in the face enumeration. We give a concise
description of how the computation can be achieved without performing
any convex hull or face lattice computation. As applications, we
deduce formulas for sparse paving matroids and rank 2
matroids. These are two families that appear in other contexts within
combinatorics.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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