This question is a variant of the question posed by Brian Hopkins.

Let pdist(n) be the number of partitions of n into distinct parts. Also, let pdist ( S, n) denote the number of partitions of n into distinct parts, all of which come from the set S. (Here again we take S to be some subset of { 1, 2, … , n }. )

Is it true that for every t from 0 to pdist(n), we can find a set S such that

pdist ( S, n) = t ?