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 Grinberg’s results [14], and interpret these in terms of the tridendriform structure of . We then present a variant of Rey’s self-dual Hopf algebra of set partitions [35] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.
Revised: 2018-02-26
Accepted: 2018-06-17
Published online: 2018-11-30
DOI: https://doi.org/10.5802/alco.28
Classification: 16T30, 05E05, 05A18
Keywords: Noncommutative symmetric functions, Quasi-symmetric functions, Bell polynomials, Dendriform algebras
@article{ALCO_2018__1_5_653_0,
author = {Novelli, Jean-Christophe and Thibon, Jean-Yves and Toumazet, Fr\'ed\'eric},
title = {Noncommutative Bell polynomials and the dual immaculate basis},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {1},
number = {5},
year = {2018},
pages = {653-676},
doi = {10.5802/alco.28},
zbl = {06987761},
mrnumber = {3887406},
language = {en},
url = {alco.centre-mersenne.org/item/ALCO_2018__1_5_653_0/}
}
Novelli, Jean-Christophe; Thibon, Jean-Yves; Toumazet, Frédéric. Noncommutative Bell polynomials and the dual immaculate basis. Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 653-676. doi : 10.5802/alco.28. https://alco.centre-mersenne.org/item/ALCO_2018__1_5_653_0/
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