I randomly came across this question today, and remembered I'd written a function to do this in Haskell not too long ago.

 distinctPartitions 0 = [[]] distinctPartitions n = [x:xs | x <- [1..n], xs <- distinctPartitions (n - x), null xs || x < head xs] 

Here's the same idea implemented in python, additionally using memoization:

from functools import lru_cache @lru_cache(maxsize=None) def distinctPartitions(n):   if n == 0:   return [[]]  return [[x] + xs for x in range(1, n+1) for xs in distinctPartitions (n-x) if xs == [] or x < xs[0]]  

My Mathematica skills are rather rusty, so I'll leave it to you to translate in case you want to try this approach - should be straightforward, hopefully.

I randomly came across this question today, and remembered I'd written a function to do this in Haskell not too long ago.

 distinctPartitions 0 = [[]] distinctPartitions n = [x:xs | x <- [1..n], xs <- distinctPartitions (n - x), null xs || x < head xs] 

Here's my (probably clumsy) attempt to translate this to Mathematica (with memoization):

distinctPartitions[0] := distinctPartitions[0] = {{}}; distinctPartitions[n_] :=    distinctPartitions[n] = Flatten[Table[  Prepend[xs, x], {x, 1, n}, {xs,    Select[distinctPartitions[n - x],    Length@# == 0 || x < First@# &]}], 1]