Nice. I wonder if there is a more direct way to sample the distribution, instead of computing the number of partitions for each of the possible first choices (this appears to be what you do, although the code is too golfed for me to really parse). For this application, you don't need the actual number of such partitions, just their ratios.

BTW, did you have in mind a cool application of uniformly sampling partitions, or was this just "because it's there"? ;)