Abstract
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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{c}_p(n,k)$$\\end{document} of xnyk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x^ny^k$$\\end{document} 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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{c}_p(n,k)$$\\end{document} 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.
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