Vol. 14, No. 5, 2021

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Catalan recursion on externally ordered bases of unit interval positroids

Jan Tracy Camacho and Anastasia Chavez

Vol. 14 (2021), No. 5, 893–905

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. A matroid is a combinatorial object that generalizes the notion of linear independence with connections to many 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.

matroid, positroid, Catalan number, Catalan recursion, poset
Mathematical Subject Classification
Primary: 05B35
Supplementary material

Code for ...

Received: 29 April 2021
Revised: 1 June 2021
Accepted: 2 June 2021
Published: 9 February 2022

Communicated by Stephan Garcia
Jan Tracy Camacho
Department of Mathematics
San Francisco State University
San Francisco, CA
United States
Anastasia Chavez
Department of Mathematics and Computer Science
Saint Mary’s College of California
Moraga, CA
United States