Catalan Recursion on Externally Ordered Bases of Unit Interval Positroids

TL;DR

This paper explores the recursive structure of externally ordered bases of unit interval positroids, revealing a Catalan-based recursion and providing algorithms for their enumeration and lattice construction.

Contribution

It introduces a new recursive framework for the bases of unit interval positroids and presents algorithms for their lattice construction and enumeration.

Findings

The poset of externally ordered UIP bases follows a Catalan recursion.

An explicit algorithm for constructing UIP lattices from lower ranks is provided.

A simple formula for enumerating UIP bases is derived.

Abstract

The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the $n^{th}$ object from previous ones. Matroids are combinatorial objects that generalize the notion of linear independence and have connections with other fields of mathematics. A family of matroids, called unit interval positroids (UIP), are Catalan objects induced by the antiadjacency matrices of unit interval orders. Associated to each UIP is the set of externally ordered bases, which due to Las Vergnas, produces a lattice after adjoining a bottom element. We study the poset of externally ordered UIP bases and the implied Catalan-induced recursion. Explicitly, we describe an algorithm for constructing the lattice of a rank $n$ UIP from the lattice of lower ranks. Using their inherent combinatorial structure, we define a simple formula to enumerate the bases for a given UIP.