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.

• Type: Journal
• Year: 2000
• Where: COMBINATORICS
• Authors: Mahendra Jani, Robert G. Rieper
• Keywords: COMBINATORICS 2000, Continued Fraction, Generating Functions, Several Catalan Problems