  Given a permutation $f$, we study the positroid Catalan number $C_f$ defined
to be the torus-equivariant Euler characteristic of the associated open
positroid variety. We introduce a class of repetition-free permutations and
show that the corresponding positroid Catalan numbers count Dyck paths avoiding
a convex subset of the rectangle. We show that any convex subset appears in
this way. Conjecturally, the associated $q,t$-polynomials coincide with the
generalized $q,t$-Catalan numbers that recently appeared in relation to the
shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky homology of
Coxeter links.
