• 1 Laboratoire Bordelais de Recherche en Informatique [LaBRI]
• 2 Institut für Mathematik [Zürich]
• 3 Laboratoire d'Informatique Gaspard-Monge

We consider the multivariate generating series $F_P$ of $P-$partitions in infinitely many variables $x_1, x_2, \ldots$ . 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 $\mathbf{WQSym}$ with a non-negative multiplication table, which lifts a basis of $\textit{QSym}$ introduced by K. Luoto.