Enumerating Restricted Dyck Paths with Context-Free Grammars

2020/09/18 by AJ Bu, Bu, AJ, Robert Dougherty-Bliss +1
Computer Science · Mathematics · #05C30 #Advanced Combinatorial Mathematics #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2009.09061

openalex publication_date 2020/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The number of Dyck paths of semilength n is famously C_n, the nth Catalan number. This fact follows after noticing that every Dyck path can be uniquely parsed according to a context-free grammar. In a recent paper, Zeilberger showed that many restricted sets of Dyck paths satisfy different, more complicated grammars, and from this derived various generating function identities. We take this further, highlighting some combinatorial results about Dyck paths obtained via grammatical proof and generalizing some of Zeilberger's grammars to infinite families.

Citations

• Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming

Related

• A new statistic on Dyck paths for counting 3-dimensional Catalan words
• Dyck Numbers, III. Enumeration and bijection with symmetric Dyck paths
• Counting generalized Dyck paths
• Restricting Dyck Paths and 312-avoiding Permutations
• The 136th manifestation of C_n