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Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras. (English) Zbl 1309.16023
Bell polynomials appear in the combinatorics of set partitions and in the composition of formal diffeomorphisms of \(\mathbb R\), or equivalently in the coproduct of the Faà di Bruno Hopf algebra. They admit a recursive definition, and explicit descriptions in terms of rooted trees or determinants. These results are here extended to a noncommutative version of these polynomials: they can be inductively defined, admit an explicit description in terms of planar rooted trees or quasideterminants. They are related to the composition of diffeomorphisms on varieties, or in a similar way appear in the coproduct of the noncommutative Dynkin Faà di Bruno Hopf algebra.
The language of incidence Hopf algebras is used in this text for the Faà di Bruno and Dynkin Faà di Bruno Hopf algebras, and a new description of the antipode of such objects is given by a quasideterminant. The text also contains a discussion on Möbius inversion in certain variants of Faà di Bruno and Dynkin Faà di Bruno Hopf algebras built from Bell polynomials.

MSC:
16T30 Connections of Hopf algebras with combinatorics
16T05 Hopf algebras and their applications
05E15 Combinatorial aspects of groups and algebras (MSC2010)
06A11 Algebraic aspects of posets
05A18 Partitions of sets
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