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

Catalan Recursion on Externally Ordered Bases of Unit Interval Positroids