[kmill]

Is the first part similar to the "W function" from Tutte's A ring in graph theory (1947)? If I remember correctly, it generalizes the chromatic/flow/Tutte polynomial to yield a linear combination of disjoint unions of bouquets (i.e., a linear combination of ordered lists of natural numbers, describing the dimensions of the cycle spaces of each connected component in the state sum expansion of the chromatic/flow/Tutte polynomial).

[ColinWright]

More abstract than that. It generated an element from an infinitely generated ring. If there was a colouring with colour sets of size x, y, and z, then it included the element P_{x-1}.P_{y-1}.P_{z-1} from the ring. Then it summed over all colourings.

So if there is a colouring of $G$ using $n$ colours with induced sets of size 4, 4, 2, and 1, then the function evaluated at $n$ included P_3^2 P_1 P_0.

(The "-1" is a technical thing that only feels obvious once you've played with it enough).

You end up with what the values should be, but the clever part was using "umbral evaluation" on the polynomial, and not just simple substitution. It might be possible to find the details on the web, but this was 35 years ago.

[kmill]

This reminds me somewhat of Stanley's symmetric chromatic function, which I guess came about ten years later (1995). Actually, what you're describing seems to be something I rediscovered a couple years ago, where P_k is interpreted as the kth power sum symmetric polynomial in n variables.

I'd be interested in learning more about your invariant. Do you have a copy of your thesis anywhere?

[ColinWright]

I might have a soft copy of the thesis, but I can only guarantee hard copy. If you're interested then the best thing to do is email me so it goes into my work queue, then I can have a quick hunt and see what I can produce easily.

Email(s) in my profile.