The multiplication of $k$-statistics is treated by Fisher , Tukey , , Wishart , Kendall , Abdel-Aty , Dwyer and Tracy , and Tracy . Various rules, procedures, functions and tables have been devised to aid in this task. The rules and procedures derived from the combinatorial properties of the $k$-statistics and other symmetric functions linearly related to them. When dealing with generalized polykas is was found convenient to ignore communtativity of multiplication in denoting the various symmetric functions and use ordered partitions to represent them rather than the usual partitions . The simplicity of the relationships among the symmetric functions, which results from the use of ordered partitions, may be used to obtain simple multiplication procedures.
E. J. Carney. "Multiplication of Polykays Using Ordered Partitions." Ann. Math. Statist. 41 (5) 1749 - 1752, October, 1970. https://doi.org/10.1214/aoms/1177696819