Consider the set $[n]$ and a positive integer $k$. Now consider the set $F$ of multisets $M$ of subsets of $[n]$ such that:
- There are exactly $k$ elements in $M$
- Subsets can appear more than once in $M$ ( So for example $M$ can contain the same subset twice)
- Each subset $A$ in $M$ has an odd number of elements
- Two intersecting subsets $A$ and $B$ in $M$ satisfy $A\subseteq B$ or $B\subseteq A$
- Every element of $[n]$ belongs to at least one of the subsets in $M$.
We call two multisets $M$ and $N$ of $F$ isomorphic if there is a bijection $\sigma$ of $[n]$ such that every subset $s$ of $[n]$ appears with the same multiplicity in $M$ as $\sigma(s)$ in $N$.
How many different isomorphism types does $F$ have?
The problem when we take $n=9$ and $k=4$ is the same as the question posed in this answer.
I have been trying to write up a recursion and use it to make a table of some values, I will probably do so in the future.