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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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