How many labelled undirected graphs are there with precisely $n$ vertices, $k$ edges, and $t$ triangle subgraphs? (By triangle I mean a graph with three vertices and three edges.)
(Clarification: I am interested only in graphs with no self loops or multiple edges. The graphs do not need to be connected.)
I would appreciate any pointers to results on this problem (exact or approximate). The results I am able to find are typically for unlabelled graphs (e.g. a count up to $n=12$), but in this case I am interested in labelled ones.
I am also interested in possible computational methods for this that are better than brute enumeration.