2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/166142We propose new algorithms for generating $k$-statistics, multivariate $k$-statistics, polykays and multivariate polykays. The resulting computational times are very fast compared with procedures existing in the literature. Such speeding up is obtained by means of a symbolic method arising from the classical umbral calculus. The classical umbral calculus is a light syntax that involves only elementary rules to managing sequences of numbers or polynomials. The cornerstone of the procedures here introduced is the connection between cumulants of a random variable and a suitable compound Poisson random variable. Such a connection holds also for multivariate random variables.A table with computational times, obtained with the forthcoming MathStatica release 2 (Colin Rose, private communication), has been added. In press Stat. CompStatistics TheoryCombinatoricsComputation65C60, 05A40, 68W30, 62H99text