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Noncommutative Bell polynomials and the dual immaculate basis. (English) Zbl 1433.16041
Summary: We define a new family of noncommutative Bell polynomials in the algebra of free quasi-symmetric functions and relate it to the dual immaculate basis of quasi-symmetric functions. We obtain noncommutative versions of D. Grinberg’s results [Can. J. Math. 69, No. 1, 21–53 (2017; Zbl 1384.05158)], and interpret these in terms of the tridendriform structure of \(\mathbf{WQSym}\). We then present a variant of M. Rey’s self-dual Hopf algebra of set partitions [“Algebraic constructions on set partitions”, in: 9th Formal Power Series and Algebraic Combinatorics (2007)] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.

MSC:
16T30 Connections of Hopf algebras with combinatorics
05E10 Combinatorial aspects of representation theory
05A18 Partitions of sets
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