I'm interested in sampling monotone increasing Boolean functions on $n$ input bits uniformly at random. I understand that this is equivalent to approximating the Dedekind numbers ($D_n = $ the number of monotone increasing Boolean functions on $n$ bits), so I don't expect a method to be very efficient, but I'm interested in a nontrivial method: not generating all of them and sampling from them and not rejection sampling.
Antichains are sets of incomparable points in a poset. The approach that seems to make sense to me would be sampling antichains in the Boolean cube: every monotone increasing Boolean function $f$ is defined by its minimal incomparable $1$s, which we will call $\partial f$ in this thread, and every set of incomparable bitstrings correspond to a monotone increasing (and decreasing) Boolean function. I've attempted to sample these, but all of my methods are not uniform, biased towards unbalanced functions.
Any thoughts on this would be appreciated; I am not necessarily expecting a full solution to be found in this thread.