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@Hugh - I hope it's finally correct - with partitions of distinct parts. For example, in the case of 5 what we have is 41 vs 5, in the case of 6 - 51 vs 321, in the case of 7 - 7 and 421 vs 61 and 43 etc.
Last time I saw the Ulam's book (a Russian translation of the 1st edition) was almost ten years ago, so I completely forgot the names he referred to, but now that you mention Cashwell and Everett, I recall it instantly that they were mentioned there. Your last link is indeed much more exciting - and indeed more likely to be relevant. It'd be quite interesting to know if there is some number-theoretic meaning of that.
You mean A/A[A,A]A, rather than A/[A,A], right? (If we mod out the linear span of commutators, this does not really make thinks commutative [and derived functors compute cyclic homology, I believe].) The derived functors of this are briefly discussed (for some examples) in arxiv.org/abs/math/0610410.
The last statement of the post (In fact, any finite associative algebra with nontrivial grading over Z_2 cannot be given a compatible Z grading.) is clearly wrong for the Grassmann algebra? You should be a bit more careful here.
