Last Symbol Distribution in Pattern Avoiding Catalan Words

We study the distribution of the last symbol statistics on the sets of Catalan words avoiding a pattern of length at most three. For each pattern p , we provide a bivariate rational generating function where the coefficient cₚ(n,k) c p ( n , k ) of xⁿyᵏ x n y k in its series expansion is the number of length n Catalan words avoiding p and ending with the symbol k . We deduce recurrence relations or closed forms for cₚ(n,k) c p ( n , k ) and we provide asymptotic approximations for the expectation of the last symbol on all Catalan words avoiding p . We end this paper by describing a computational approach using computer algebra.