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 \(\varvec{c}_p(n,k)\) ${\mathit{c}}_p(n,k)$ of \(x^ny^k\) $x^n{y}^k$ 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 \(\varvec{c}_p(n,k)\) ${\mathit{c}}_p(n,k)$ 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.