Modular Fuss-Catalan numbers

The modular Catalan numbers $C_{k,n}$, introduced by Hein and Huang in 2016 count equivalence classes of parenthesizations of $x_0 * x_1 * \dots *x_n$ 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 $C_{1,n} = 1$ and the size of the unique class is given by the Catalan number $C_n$. In this paper we introduce modular Fuss-Catalan numbers $C_{k,n}^{m}$ which count equivalence classes of parenthesizations of $x_0 * x_1 * \dots *x_n$ where $*$ is an $m$-ary $k$-associative operation for $m \geq 2$. Our main results are a closed formula for $C_{k,n}^{m}$ and a characterisation of $k$-associativity.