A new formula for cumulants was given by Good (1975). A short proof is now given and the result is used to obtain a new formula for $k$-statistics. This formula can be used both for deriving the expressions of $k$-statistics in terms of power sums of the observations, and for checking and locating errors in formulae that are already in the literature.
"A New Formula for $k$-Statistics." Ann. Statist. 5 (1) 224 - 228, January, 1977. https://doi.org/10.1214/aos/1176343757