The enriched $q$-monomial basis of the quasisymmetric functions

We construct a new family ${\left({\eta }_{\alpha }^{\left(q\right)}\right)}_{\alpha \in \mathrm{Comp}}$ of quasisymmetric functions for each element $q$ of the base ring. We call them the “enriched $q$-monomial quasisymmetric functions”. When $r:=q+1$ is invertible, this family is a basis of $\mathrm{QSym}$. It generalizes Hoffman’s “essential quasi-symmetric functions” (obtained for $q=0$) and Hsiao’s “monomial peak functions” (obtained for $q=1$), but also includes the monomial quasisymmetric functions as a limiting case. We describe these functions ${\eta }_{\alpha }^{\left(q\right)}$ by several formulas, and compute their products, coproducts and antipodes. The product expansion is given by an exotic variant of the shuffle product which we call the “stufufuffle product” due to its ability to pick several consecutive entries from each composition. This “stufufuffle product” has previously appeared in recent work by Bouillot, Novelli and Thibon, generalizing the “block shuffle product” from the theory of multizeta values.