A partition of an integer $n$ is a non-decreasing list of positive integers summing to $n$. For example, $3$ can be partitioned as $1 + 1 + 1$, $1 + 2$ or just $3$, but $2 + 1$ is indistinct from $1 + 2$ (e.g., order does not matter).
A refinement of a partition $P$ of $n$ is (I hope I'm using this term correctly) another partition of $n$ obtained by further partitioning the elements of $P$. That definition is a little loose, but I hope an example will clarify. The partitions $(1,1,1,3)$ and $(1,1,2,2)$ both refine $(3,3)$, since they are each obtained by partitioning the elements of $(3,3)$. By contrast, $(2,4)$ is also a partition of $6$, but it does not refine $(3,3)$. (Notice refinement induces a poset structure on the collection of partitions of an integer.)
Given a particular partition $P$ of $n$, is there a built-in function in Sage to iterate over all refinements of $P$? For example, iterating over the refinements of $(2,3)$ should produce something like the list $(2,3)$, $(1,2,2)$, $(1,1,3)$, $(1,1,1,2)$, $(1,1,1,1,1)$.
I see how one could build such a function "from scratch" (list all partitions for each entry and just stitch them together), but I wonder if there is something hidden in a number theory or lattice/poset theory package that I'm not aware of.
Alternate Goal: For my purposes, I would even be happy with the ability to efficiently check whether one integer partition refines another (as opposed to returning the list of all such refinements).
Update: Some evidence that this problem is hard in general and a suggested solution via integer linear programming.