Modular Fuss-Catalan numbers

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 C m k,n 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 C m k,n , and a characterisation of k-associativity.