On generalised catalan numbers

The Catalan number C_n is defined to be $\text{2n}\text{n}\text{(n+1)}$. One of its occurrences is as the number of ways of bracketing a product of n+1 terms taken from a set with binary operation. In this note the corresponding result for a set with a k-ary operation is considered. A combinatorial proof is given which does not involve generating functions or inversion formulae. The result is further generalised to obtain a simpler proof of a formula of Erdelyi and Etherington 2], interpreted here as a result concerning a set with several k_i-ary operations.