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

TL;DR

This paper introduces the Young matroid, extending the Catalan matroid by defining bases through standard Young tableaux, incorporating multisets to generalize the combinatorial structure.

Contribution

It presents the Young matroid, a novel multiset extension of the Catalan matroid based on Young tableaux, expanding the framework of lattice path and shifted matroids.

Findings

Defines the Young matroid using Young tableaux

Extends the concept of bases to multisets

Provides a new combinatorial structure for matroids

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.