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
-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-
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.
