Graded rings with compatible S_n actions

Steven Sam and Andrew Snowden and their other collaborators call these `twisted commutative algebras' and have been having fun writing papers about properties of generators and similar.

But predating this, any topologist of a certain sort would call this a monoid in the category of symmetric abelian groups. If you do the same construction in simplicial sets (or topological spaces), an example is the sphere spectrum, with nth space $S^n$. Modules over this object are what are called symmetric spectra, and serve as one of the main models of modern day stable homotopy theory. (This was an observation by Jeff Smith in the mid 1990s.)

As a warmup, an $\mathbb{N}$-graded ring is a monoid object in the symmetric monoidal category of $\mathbb{N}$-graded abelian groups under the convolution tensor product, which you can think of as Day convolution from the usual addition on $\mathbb{N}$.

Similarly, this thing is a monoid object in the symmetric monoidal category of species in abelian groups (presheaves on the category $S$ of finite sets and bijections valued in abelian groups) under the convolution tensor product, which you can again think of as Day convolution from disjoint union on $S$.

So I might be inclined to call such a thing an $S$-graded ring, and you can feel free to replace $S$ with your favorite name for $S$, maybe $\text{FinSet}^{\times}$ or something.