Modular Catalan Numbers

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 $C_n$ enumerates parenthesizations of $x_0*\dotsb*x_n$ where $*$ is a binary operation. We introduce the modular Catalan number $C_{k,n}$ to count equivalence classes of parenthesizations of $x_0*\dotsb*x_n$ when $*$ satisfies a $k$-associative law generalizing the usual associativity. This leads to a study of restricted families of Catalan objects enumerated by $C_{k,n}$ with emphasis on binary trees, plane trees, and Dyck paths, each avoiding certain patterns. We give closed formulas for $C_{k,n}$ with two different proofs. For each $n\ge0$ we compute the largest size of $k$-associative equivalence classes and show that the number of classes with this size is a Catalan number.