Continued fractions and Catalan problems

2000/01/16 by Mahendra Jani, Jani, Mahendra, Robert G. Rieper +1
Mathematics · #05A #05C30 #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Mathematical functions and polynomials #Mathematics and Applications #math.CO #msc:05A #msc:05C30

paper · pdf · doi:10.48550/arxiv.math/0001091

9 pages, 1 figure

arxiv created 2000/01/16 · openalex publication_date 2000/01/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Patterns and Fractions
• (p,q,t)-Catalan continued fractions, gamma expansions and pattern avoidances
• Continued fractions and generalized patterns
• A derivation of the Catalan numbers from a bijection between permutations and labeled trees
• Enumerations of Permutations Simultaneously Avoiding a Vincular and a\n Covincular Pattern of Length 3