Noncommutative Bell polynomials and the dual immaculate basis

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 these 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.

The Stylic Monoid

“…The proof is nontrivial, but we followed the classical case (skew diagrams with a hole [23]), as is shown in Sagan's book [21], with the help of Fomin's growth diagrams, which may be extended to our case: partitions are replaced by compositions, appropriately ordered. We use a notion that appeared previously in the literature: composition tableaux of [10,14] (with one condition removed), and more precisely, standard immaculate tableaux [2] (see also [3], [7], [9], [1], and [18]).…”