Consider the following construction,
$$\sum_{\rho'\in\mathcal{P}(n), \rho'>\rho} f(\rho)$$
where $\rho$ and $\rho'$ are partitions of the set $\{1,2,\dots,n\}$ and $\mathcal{P}(n)$ is the set of all possible partitions.
Here, I would like to know the meaning of $\rho' > \rho$. Can anyone give me a hint? I interpret:
$\rho'$ in the set $\mathcal{P}(n)$ such that $\rho'$ is greater than $\rho$.
Does it mean I should choose $\rho'$ that has more elements than $\rho$?
It means $\rho'$ is finer than $\rho$ in the sense that every set in $\rho$ is a union of some sets from $\rho'$.
For example $\{1,\{2,3,..,n\}\}$ and $\{1, \{2\}, \{3,4,..,n\}\}$ are two partitions of $\{1,2,3,..,n\}$ and the second one is finer than the first.