I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., both over $\mathbb Z$). See Victor Reiner, *Hopf algebras in combinatorics*, chapters 2 and 5, respectively, for the definitions.

It is known that one can define a *plethysm* $f\circ g \in \mathrm{QSym}$ for any $f\in\Lambda$ and any $g\in\mathrm{QSym}$. This is described, e. g., in Claudia Malvenuto, Christophe Reutenauer, *Plethysm and conjugation of quasi-symmetric functions*, Discrete Mathematics 193, 225-233 (1998). In a nutshell, if $g$ is a sum of monomials in the $x_1$, $x_2$, $x_3$, ..., one can construct $f\circ g$ by substituting these monomials as indeterminates into $f$. It takes some more work (and is less intuitive) to define $f\circ g$ when $g$ has negative coefficients, but the above should give some feeling for what $f\circ g$ is. Notice that $e_1\circ g = g$ (where $e_1$ is the $1$-st elementary symmetric function). For $g \in \Lambda$, the plethysm $f\circ g$ becomes the usual plethysm in $\Lambda$.

In section 3 of the preprint Michiel Hazewinkel, *Explicit generators for the ring of quasisymmetric functions over the integers*, it is claimed that this construction extends to all $f\in \mathrm{QSym}$, where the monomials are substituted into $f$ in lexicographic order. I cannot follow this claim, because it seems to me that the "addition formula"

**(1)** $f \circ \left(g+h\right) = \sum\limits_{(f)} \left(f_{(1)}\circ g\right) \left(f_{(2)}\circ h\right)$ (using Sweedler notation, where $\sum\limits_{(f)} f_{(1)} \otimes f_{(2)}$ is the first coproduct of $f$)

is no longer satisfied for general $f\in\mathrm{QSym}$, and the definition of $f\circ g$ outside the case of $g$ being a sum of monomials hinges on this formula (of course, there are better definitions in the $f \in \Lambda$ case which don't depend on this formula, but they don't look generalizable at all).

What I want to know, apart from whether or not my doubts on this definition are justified, is whether there is any reasonable definition of a plethysm of two elements of $\mathrm{QSym}$ known, or whether there are good reasons no such beast exists in nature.

[**EDIT:** At a second glance, if we take Hazewinkel literally, he isn't claiming this all; he is only defining $f\circ g$ for $f$ and $g$ being monomial quasisymmetric functions, which (I guess) he can do as he pleases. But I think he is trying to define $f\circ g$ for all $f\in\mathrm{QSym}$ and $g\in\mathrm{QSym}$. If I interpret his definition of $f\circ g$ as being only formulated for the monomial quasisymmetric functions, and then try to extend it using **(1)** to all $f\in\mathrm{QSym}$ and $g\in\mathrm{QSym}$, then I think I obtain a contradiction due to the non-cocommutativity of $\mathrm{QSym}$.]

[Let me remark that Hazewinkel's proof of the polynomial freeness of $\mathrm{QSym}$ does not depend on this kind of plethysm. He only ever uses it for $f\in\Lambda$. A clean version of his proof can be found in Chapter 6 of Michiel Hazewinkel, Nadiya Gubareni, V.V. Kirichenko, *Algebras, Rings and Modules, Volume 3*, and the only gap in it (the unproven footnote 13) can be filled in using Section 2 of David E. Radford, *A natural ring basis for the shuffle algebra and an application to group schemes*.]

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