  We 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.
