  Catalan numbers arise in many enumerative contexts as the counting sequence
of combinatorial structures. In this work, we consider natural Markov chains on
some of the realizations of the Catalan sequence. While our main result is in
deriving an $O(n^2 \log n)$ bound on the mixing time in $L_2$ (and hence total
variation) distance for the random transposition chain on Dyck paths, we raise
several open questions, including the optimality of the above bound. The
novelty in our proof is in establishing a certain negative correlation property
among random bases of lattice path matroids, including the so-called Catalan
matroid which can be defined using Dyck paths.
