This was originally an answer, but didn't work out; consider it a suggestion. The following is taken from a set of notes on combinatorics by Stanley I can no longer find online.

Let $B_m$ denote the poset of subsets of $\{ 1, 2, ... m \}$. Given a group $G$ acting on $\{ 1, 2, ... m \}$, let $B_m/G$ denote the quotient poset whose elements are orbits of the action of $G$ and where $x \le y$ if some element in the orbit $x$ is contained in some element of the orbit $y$. It is not hard to see that like $B_m$, the quotient $B_m/G$ is graded and rank-symmetric, with rank given by the size of the corresponding subset.

**Theorem:** $B_m/G$ is unimodal (the size of the ranks increase, then decrease).

The proof is somewhat involved; one defines certain linear operators between the free vector spaces on each rank. According to Stanley there is no known proof without linear algebra.

It seems plausible that the poset of distinct partitions into parts of size at most $n$ might be of the form $B_m/G$ for some $G$, but I have not been able to find such a construction.