Modular Fuss-Catalan numbers

Abstract

The modular Catalan numbers C k,n , introduced by Hein and Huang in 2016 count equivalence classes of parenthesizations of x0 * x1 * • • • * xn where * is a binary k-associative operation and k is a positive integer. The classical notion of associativity is just 1-associativity, in which case C1,n = 1 and the size of the unique class is given by the Catalan number Cn. In this paper we introduce modular Fuss-Catalan numbers C m k,n which count equivalence classes of parenthesizations of x0 * x1 * • • • * xn where * is an m-ary k-associative operation for m ≥ 2. Our main results are a closed formula for C m k,n and a characterisation of k-associativity.