Wishart has demonstrated and Kendall has justified the application of a combinatorial method to products of $k$-statistics. Wishart's combinatorial method is, essentially, that introduced by Fisher though modifications were necessary since Fisher's method is applied directly to the writing of cumulant results while Wishart's method applies to products of $k$-statistics. It is the purpose of this paper to show that the combinatorial method may be further modified and extended so as to produce products of polykays and to present general formulae resulting from the new combinatorial method.
"A Combinatorial Method for Products of Two Polykays with Some General Formulae." Ann. Math. Statist. 35 (3) 1174 - 1185, September, 1964. https://doi.org/10.1214/aoms/1177703275