Catalan matroid decompositions of certain positroids

TL;DR

This paper explores a specific class of positroids derived from permutations, providing formulas for their bases and Tutte polynomials using Catalan numbers, and revealing their structure as unions of simpler matroids.

Contribution

It introduces a novel decomposition of certain positroids into unions of Catalan and trivial matroids, with explicit enumeration formulas.

Findings

Enumeration of bases as sums of Catalan number products

Explicit formulas for Tutte polynomials of these positroids

Structural decomposition into disjoint unions of simpler matroids

Abstract

A positroid is the matroid of a matrix whose maximal minors are all nonnegative. Given a permutation $w$ in $S_n$, the matroid of a generic $n \times n$ matrix whose non-zero entries in row $i$ lie in columns $w(i)$ through $n+i$ is an example of a positroid. We enumerate the bases of such a positroid as a sum of certain products of Catalan numbers, each term indexed by the $123$-avoiding permutations above $w$ in Bruhat order. We also give a similar sum formula for their Tutte polynomials. These are both avatars of a structural result writing such a positroid as a disjoint union of matroids, each isomorphic to a direct sum of Catalan matroids and a matroid with one basis.