Inspired by this question about defining the partition function on non integers, I was thinking about what sorts of other objects can a partition function be defined on.
I noticed that if we have an undirected unlabeled graph $G$ that consists of just a long chain of $n$ nodes connected by $n-1$ edges. Then the number of distinct subgraphs of $G$ that can be formed by deleting edges is simply equal to $p(n)$ the original partition function.
It's then natural to ask about the function $p(G)$ which takes a finite graph as an input and outputs the number of distinct subgraphs of $G$ that can be formed by deleting edges.
Is anything known about these partitions? Do we have efficient ways of computing this? Is there a way to lift for example the pentagonal number theorem to the graph setting or even Ramanujan congruences to the graph setting?
A slightly different definition:
If we instead consider a graph $G$ with vertices $V = v_1, ... v_n$ we can define a partition of $G$ as creating sets of vertices $V_1, ... V_r$ which distinctly partition the set $V$ and then this induces a subgraph of $G$ where we delete any edge between distinct $V_i, V_j$ but do not delete edges that are internal to a $V_k$. This definition is a bit more complex making it less desirable but it also embodies the spirit of partitioning a bit better. I would be interested in this as well.