A partition of $p$ of $n$ consists of a sum to $n$. We say $p$ dominates $q$ if the summands in $q$ are achievable by those in $p$. So for example, $(6,5)$ dominates $(5,5,1)$ but not $(7,4)$. $(n)$ dominates everything, $(1,...,1)$ dominates nothing.

We now add up all partitions and their dominatories, and call this function $F$. Are there known asymptotics on $F$?

Motivation is from looking at multinomials.