The preprint https://arxiv.org/abs/1701.01314 of Magnus Carlson, "Classification of plethories in characteristic zero" answers the question about plethories over $\mathbb{Q}$ in the affirmative.


As far as I know, this question is still open. The only result in this direction I'm aware of is the subject of a paper by Buium called "Arithmetic analogues of derivations" where he gives a complete classification of biring structures on the polynomial algebra in two variables. He doesn't even restrict to $\mathbf{Q}$ coefficients. The answer is that you ...


As per @darij's suggestion regarding my above comment: Loehr and Remmel's A computational and combinatorial exposé of plethystic calculus 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.

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