Modular Catalan numbers

The Catalan number C-n enumerates parenthesizations of x(0) *. . . *x(n) where * is a binary operation. We introduce the modular Catalan number C-k,C-n to count equivalence classes of parenthesizations of x(0) * . . . * 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 Ck, with emphasis on binary trees, plane trees, and Dyck paths, each avoiding certain patterns. We give closed formulas for Ck,n with two different proofs. For each n >= 0 we compute the largest size of k-associative equivalence classes and show that the number of classes with this size is a Catalan number. (C) 2016 Elsevier Ltd. All rights reserved.