  We construct a new family $\left( \eta_{\alpha}^{\left( q\right) }\right)
_{\alpha\in\operatorname*{Comp}}$ 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
$\operatorname{QSym}$. 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.

Combinatorics