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$$r$$-Bell polynomials in combinatorial Hopf algebras. (Polynomes de $$r$$-Bell dans les algèbres de Hopf combinatoires.) (English. French summary) Zbl 1370.16030
Partial multivariate Bell polynomials were introduced in [E. T. Bell, Ann. Math. (2) 35, No. 2, 258–277 (1934; Zbl 0009.21202)] and are known to be related to, for example, Stirling numbers (of both the first and second kind). These polynomials were generalized to $$r$$-Bell polynomials in [M. Mihoubi and M. Rahmani, “The partial $$r$$-Bell polynomials”, Preprint, arXiv:1308.0863].
In this work, the authors define three versions of the $$r$$-Bell polynomials in three different combinatorial Hopf algebras using the same formula. The combinatorial Hopf algebras are $$\mathbf{Sym}^{(2)}$$, the algebra of bisymmetric functions, $$\mathbf{NCSF}^{(2)}$$, the algebra of noncommutative bisymmetric functions, and $$\mathbf{WSym}^{(2)}$$, the algebra of $$2$$-colored word symmetric functions. Specifically, for each of the combinatorial Hopf algebras above, there is a unique derivation $$\partial$$ which satisfies specific properties based on the underlying combinatorial Hopf algebra. The $$r$$-Bell polynomial $$B_{n+r,k+r}^r$$ is defined to be the coefficient of $$t^k$$ in the expansion of $$a_1^r(tb_1 + \partial)^n$$, where $$(a_i)_{i\geq1}$$ and $$(b_i)_{i\geq 1}$$ represent specific generators for each algebra. A characterization of each version of $$B_{n+r,k+r}^r$$ is given in terms of bicolored set partitions.

MSC:
 16T30 Connections of Hopf algebras with combinatorics 05A18 Partitions of sets 11B73 Bell and Stirling numbers
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References:
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