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Thanks @Dietrich, so you're not aware of any other results regarding polynomials which are "killed" by certain permutations? I sort of wonder if this is connected to the Hodge decomposition and filtration that Loday gives for Hochschild (co)homology.
@YemonChoi So, it's living in something like the cobar construction on $R[x]$, or you can think of it as being in $H^*(\mathbb{G}_a,\mathbb{G}_a)$ where the first is a monoid acting trivially on the module on the right, as described by Demazure and Gabriel. I don't really know of another way to place polynomial cocycles inside of some Hochschild cohomology.
Wow @Lubin I'd love to talk to you more about this if you'd be willing. It specifically concerns the cohomology theory you and John Tate describe the first two degrees of in your 1966 paper, specifically, the third degree of that cohomology theory. I am essentially trying to reprove a theorem of Lazard's from a purely homological standpoint.
Ping me in the homotopy-theory chat if you want the mathematica notebook I wrote to compute these things for finite degrees. Or if you just want to see the polynomials.
FWIW I have completely determined this power series up to degree 5, and there is definitely a pattern (although the first 3 degrees are 0 if $f$ is commutative).