Catalan Matroid Decompositions of Certain Positroids
Tuesday, March 10, 2015 - 2:00pm - 3:00pm
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 3$-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 adisjoint union of a direct sum of Catalan matroids (up to isomorphism) and free matroids.