My comment makes no sense! Repetition implies that $d(S) = 0$. I'm going to have to stop and think about what I am asking.

Hi Peter, I don't think this is correct. With ''m-subsets of $\{1,2,\cdots,n\}$'' you are not allowing repetition, but I mean for the question to allow repetition. This is wholly my fault, I should never have called $S$ a set!

This question might work better on stats.stackexchange.com

You might be able to get a reasonable solution using loopy belief propagation' approaches (its sometimes also called generalized belief propagation', see en.wikipedia.org/wiki/Belief_propagation). As far as I know, these are approaches to solving dynamic programing type problems when the graphs have cycles.

mmm I wonder if there is a relationship between ease' of spigot algorithm and goodness' of equidistribution?

Perhaps people on stack overflow would be able to help you out on this one too?

I doubt it. I personally am much happier believing a published proof (that I can't find an error in) than the output of some sort of RNG built by hand in the real world.

but +1 anyway because I think this formula is really cool.

see Steve's comment.

Nice !