it seems like the process of taking the zariski tangent space of a schubert variety at the basepoint in the a-series case gives us a map from n! (the schubert varieties on the flag variety) to catalan(n) (the invariant distributions on the flag variety), so i started trying to connect this with other situations where there's a map from n! to catalan(n) ...

(it's not obvious to me yet whether the map that we're getting is surjective in general, though i'm guessing that it probably is.)

a permutation of the spaces between a string of factors gives a "bracketing" of the string, by using the permutation to determine the "precedence" of the joining operators corresponding to the spaces. (moreover, this process might play a significant role in certain aspects of higher-dimensional category theory...) so i decided to check whether permutations sharing a bracketing are essentially the same as schubert varieties sharing a zariski tangent space. and the answer that i seem to be getting so far is no.

that's somewhat of a disappointment, but it reminded me of a similar disappointment that i think i vaguely remember... which raises the possibility of trying to get those two disappointments to "cancel out"...


so let's consider the action of the group n! on the subsets of n.... ??....