Given a (bounded affine) permutation
f
f
, we study the positroid Catalan number
C
f
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
q,t
-polynomials coincide with the generalized
q
,
t
q,t
-Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov–Rozansky homology of Coxeter links.