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Continued fractions and Catalan problems

Mahendra Jani·Robert G. Rieper
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Abstract

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.

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Mahendra Jani, Robert G. Rieper. 2000-01-16. Continued fractions and Catalan problems. https://arxiv.org/abs/math/0001091

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