That diagram of partitions of a set with 5 elements i first guessed would be 256=2^5. (but you are 'coarse graining ' those--similar to difference between identical and indistinguishable particles and identical and distinguishable particle--seperates bose-einstein from boltzmannn statistics).
But those are 'Bell numbers' (B(5)=52 -reminds me of a b52--a band and a plane). I'm not sure there is a 'closed form solution' except as an approximation. You compute those recursively--need a calculator or computer.