Let $p$ and $q$ be partitions of $n$. We say $q$ refines $p$ if the parts of $p$ can be subdivided to produce the parts of $q$. For example, $(5,5,1)$ refines $(6,5)$ but not $(7,4)$. $(n)$ refines only itself, and $(1,...,1)$ refines all partitions of $n$.

For each partition of $n$, count the number of partitions refining it. Let $F(n)$ be the sum of these counts. For example, $F(3) = 3+2+1=6$, and $F(4) = 5+3+3+2+1=14$. What is known about the asymptotics of $F(n)$?

My motivation is from looking at multinomials.

Note: Refinement is not the dominance order. It is related to the refinement of set partitions.

EDIT: I found why I was looking at these numbers: