Super quasi-symmetric functions via Young diagrams

TL;DR

This paper introduces a new construction linking multivariate generating series of P-partitions with an analog involving two infinite alphabets, leading to a basis of WQSym with positive structure constants.

Contribution

It provides an explicit map from the generating series of P-partitions to an analog with two alphabets and constructs a basis of WQSym with non-negative multiplication.

Findings

Established a bijective map between $F_P$ and $N_P$

Constructed a basis of WQSym with positive structure constants

Extended the basis of QSym by Luoto to a noncommutative setting

Abstract

We consider the multivariate generating series $F_P$ of $P$-partitions in infinitely many variables $x_1, x_2 , \dots$. For some family of ranked posets $P$, it is natural to consider an analog $N_P$ with two infinite alphabets. When we collapse these two alphabets, we trivially recover $F_P$. Our main result is the converse, that is, the explicit construction of a map sending back $F_P$ onto $N_P$. We also give a noncommutative analog of the latter. An application is the construction of a basis of WQSym with a non-negative multiplication table, which lifts a basis of QSym introduced by K. Luoto.