Word Bell Polynomials

Partial multivariate Bell polynomials have been de ned by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities etc. Many of the formulæ on Bell polynomials involve combinatorial objects (set partitions, set partitions into lists, permutations etc). So it seems natural to investigate analogous formulæ in some combinatorial Hopf algebras with bases indexed by these objects. In this paper we investigate the connexions between Bell polynomials and several combinatorial Hopf algebras: the Hopf algebra of symmetric functions, the Faà di Bruno algebra, the Hopf algebra of word symmetric functions etc. We show that Bell polynomials can be de ned in all these algebras and we give analogues of classical results. To this aim, we construct and study a family of combinatorial Hopf algebras whose bases are indexed by colored set partitions.

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