Modular Catalan Numbers
Nickolas Hein, Jia Huang

TL;DR
This paper introduces the modular Catalan numbers $C_{k,n}$, which count equivalence classes of parenthesizations under a $k$-associative law, extending classical Catalan combinatorics with new formulas and structural insights.
Contribution
It defines and analyzes the modular Catalan numbers $C_{k,n}$, providing closed formulas and studying their combinatorial structures and enumerations.
Findings
Closed formulas for $C_{k,n}$ derived
Largest size of $k$-associative classes computed
Number of largest classes equals a Catalan number
Abstract
The Catalan number enumerates parenthesizations of where is a binary operation. We introduce the modular Catalan number to count equivalence classes of parenthesizations of when satisfies a -associative law generalizing the usual associativity. This leads to a study of restricted families of Catalan objects enumerated by with emphasis on binary trees, plane trees, and Dyck paths, each avoiding certain patterns. We give closed formulas for with two different proofs. For each we compute the largest size of -associative equivalence classes and show that the number of classes with this size is a Catalan number.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Coding theory and cryptography
