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Super quasi-symmetric functions via Young diagrams

Abstract : 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.
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https://hal.inria.fr/hal-01207580
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Jean-Christophe Aval, Valentin Féray, Jean-Christophe Novelli, Jean-Yves Thibon. Super quasi-symmetric functions via Young diagrams. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.169-180, ⟨10.46298/dmtcs.2390⟩. ⟨hal-01207580⟩

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