Modular Catalan Numbers

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.