  A positroid is the matroid of a matrix whose maximal minors are all
nonnegative. 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 $123$-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 a disjoint union of matroids,
each isomorphic to a direct sum of Catalan matroids and a matroid with one
basis.
