Noncommutative Bell polynomials and the dual immaculate basis

TL;DR

This paper introduces noncommutative Bell polynomials within free quasi-symmetric functions, linking them to the dual immaculate basis, and explores their algebraic structures and equivalence classes.

Contribution

It defines a new family of noncommutative Bell polynomials and relates them to existing bases and algebraic structures, extending prior results to a noncommutative setting.

Findings

Established noncommutative Bell polynomials in free quasi-symmetric functions.

Connected these polynomials to the dual immaculate basis.

Described Bell equivalence classes via poset linear extensions.

Abstract

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 [Canad. J. Math. 69 (2017), 21--53], and interpret them in terms of the tridendriform structure of WQSym. We then present a variant of Rey's self-dual Hopf algebra of set partitions [FPSAC'07, Tianjin] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.