Summary: Multivariate partial Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities, etc. Many of the formulas on Bell polynomials involve combinatorial objects (set partitions, set partitions into lists, permutations, etc.). So it seems natural to investigate analogous formulas in some combinatorial Hopf algebras with bases indexed by these objects. In this paper we investigate the connections 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 defined in all these algebras, and we give analogs of classical results. To this aim, we construct and study a family of combinatorial Hopf algebras whose bases are indexed by colored set partitions.
Bell polynomials, munthe-kaas polynomials, set partitions, colored set partitions, Hopf algebras, symmetric functions, word symmetric functions, faà di bruno algebra, Lagrange inversion