The Young matroid: A multiset extension of the Catalan matroid to arbitrary Young diagrams

Abstract

Introduced by Ardila (J. Combin. Theory Ser. A, 2003), the Catalan matroid is obtained by defining the bases of the matroid using Dyck paths from $(0,0)$ to $(n,n)$. Further research has gone into the topic, with variants like lattice path matroids (introduced by Bonin, de Mier, and Noy (J. Combin. Theory Ser. A, 2003)) and shifted matroids (introduced independently by Klivans (2003), and Ardila) being studied intensively. In this short note, we introduce the Young matroid, an extension of the Catalan matroid, where the bases are defined using the standard Young tableaux of a fixed shape. This extension necessarily involves the consideration of independent multisets and multiset bases.

Section 2 has outlines of all proofs, and not complete proofs. While completing the proof of Theorem 1.2, an error was found which cannot be fixed. In fact, The claim of Theorem 1.2 was later found to be incorrect: a counterexample was found which was an integer polytope but not a matroid basis polytope

References in corpus (1)

• Lattice Path Matroids: Negative Correlation and Fast Mixing