Continued Fractions and Catalan Problems
Copyright policy )Continued Fractions and Catalan Problems
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of $(132)$-pattern avoiding permutations that have a given number of increasing patterns of length $k$. This extends and illuminates a result of Robertson, Wilf and Zeilberger for the case $k=3$.
05A; 05C30, Continued fractions, Other combinatorial number theory, Exact enumeration problems, generating functions, 05A, FOS: Mathematics, Mathematics - Combinatorics, 05C30, Combinatorics (math.CO)
05A; 05C30, Continued fractions, Other combinatorial number theory, Exact enumeration problems, generating functions, 05A, FOS: Mathematics, Mathematics - Combinatorics, 05C30, Combinatorics (math.CO)
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