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*](http://www.math.umn.edu/~reiner/Classes/HopfComb.pdf), 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)](http://www.mat.uniroma1.it/people/malvenuto/Plethysm.pdf). 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*](http://oai.cwi.nl/oai/asset/10350/10350B.pdf), 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*](http://www.sciencedirect.com/science/article/pii/0021869379901716).]



Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined?
As per @darij's suggestion regarding my above comment: Loehr and Remmel's  [A computational and combinatorial exposé of plethystic calculus](http://link.springer.com/article/10.1007%2Fs10801-010-0238-4) is a good resource for a combinatorial point of view of plethysm; I think it clarifies what one should do when both functions are quasisymmetric. **Edit:** Apparently the paper is Open Access.


