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 ?

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

Let $\operatorname{pdist}(n)$ be the number of partitions of $n$ into distinct parts. Also, let $\operatorname{pdist}(S, n)$ 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 $[n]= \{ 1, 2, ... , n \}$. )

Is it true that for every $t$ from $0$ to $\operatorname{pdist}(n)$, we can find a set $S$ such that $\operatorname{pdist} ( S, n) = t$ ?