Descent distribution on Catalan words avoiding a pattern of length at most three

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern p we provide a bivariate generating function where the coefficient of xnyk in its series expansion is the number of length np-avoiding Catalan words with k descents. As a byproduct, we enumerate the set of Catalan words avoiding p, and we provide the popularity of descents on this set.