A q-Deformation of Enriched P-Partitions

We introduce a q-deformation that generalises in a single framework previous works on classical and enriched P-partitions. In particular, we build a new family of power series with a parameter q that interpolates between Gessel’s fundamental (q = 0) and Stembridge’s peak quasisymmetric functions (q = 1) and show that it is a basis of QSym when q ∉ {−1, 1}. Furthermore we build their corresponding monomial bases parametrised with q that cover our previous work on enriched monomials and the essential quasisymmetric functions of Hoffman.