TITLE: Counting refinements of partitions
QUESTION [6 upvotes]: 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:
JonMarkPerryBlog:WordPress

REPLY [3 votes]: I assume you are asking for the number of pairs $(\pi,\sigma)$ of partitions of $n$ so that $\sigma$ is a refinement of $\pi$. Call this $q(n)$.
Let $p(n)$ be the number of partitions of $n$. $p(n) \sim \frac{1}{4\sqrt {3}~n} \exp \left( \pi \sqrt{\frac{2n}{3}}\right)$ by Hardy and Ramanujan, so $p(n)$ has upper and lower bounds of the form $\exp (c\sqrt{n})$.
$p(n) \le q(n) \le p(n)^2$ so $q(n)$ also has upper and lower bounds of the form $\exp (c \sqrt{n})$. It is natural to try to find the limit of $\frac{1}{\sqrt{n}}\log q(n)$. The $\liminf$ is at least $\pi \sqrt{2/3}$ and the $\limsup$ is at most $\pi\sqrt{8/3}$.
Here is a construction showing that the $\liminf$ is at least $\pi \sqrt{4/3}$: Consider pairs of partitions $(\pi^-,\sigma^-)$ of $n/2$. To $\pi^-$, add $n/2$ to the largest part to get a partition $\pi$ of $n$. To $\sigma^-$, add $n/2$ $1$s to get a partition $\sigma$ of $n$. No pair $(\pi,\sigma)$ is hit twice, and $\sigma$ is a refinement of $\pi$.