Modular Fuss-Catalan numbers

Abstract

The modular Catalan numbers Ck,n, introduced by Hein and Huang in 2016 count equivalence classes of parenthesizations of x0⁎…⁎xn, where ⁎ is a binary k-associative operation and k is a positive integer. The classical notion of associativity coincides with 1-associativity, in which case C1,n=1 and the single 1-equivalence class has size given by the Catalan number Cn. In this paper we introduce modular Fuss-Catalan numbers Ck,nm which count k-equivalence classes of parenthesizations of x0⁎…⁎xn where ⁎ is an m-ary k-associative operation for m≥2. Our main results are, an explicit formula for Ck,nm, and a characterisation of k-associativity.