Multivariate extensions and combinatorial interpretations of Bell polynomials are the subject of this work.
There are four papers on this subject:
Summary. Stirling and Bell polynomials, associated in one dimension to either set partitions or derivatives of a function composition, are extended to several dimensions. Basic recurrences for these polynomials are established. A combinatorial interpretation is provided in terms of colored partitions.
Discrete Mathematics, 204, 155-162, 1999.
Moments and cummulants of Poisson processes are known to involve summations over partitions of a set. Identities which generalize such formulae, from characteristic functions to arbitrary series, are presented in this paper. A probabilistic interpretation for the quotient of two such expressions is shown to represent the chance that a player is ruined in a series of games.
Transactions of the AMS, 2, 503-520, 1996.
Summary. A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.
SIAM J. Discrete Math., 2, 194-202, 1994.
Summary. The notion of compound nonhomogeneous Poisson processes provides a general framework for Bell polynomials and set partitions. Identities that can be viewed as generalizations of Dobinski's formula are obtained by interpreting them as moments of such processes.