I am teaching a combinatorics class in which I introduced the notion of a "mass formula". My terminology is inspired by the [Smith–Minkowski–Siegel mass formula](http://en.wikipedia.org/wiki/Smith%25E2%2580%2593Minkowski%25E2%2580%2593Siegel_mass_formula) for the total mass of positive-definite quadratic forms of a given size and genus. That famous mass formula is much too fancy of an example for my class. All that I really do is define the concept of the "mass" of a combinatorial object to be $1/|G|$ if $G$ is its automorphism group, and then argue that it can be easier to find the total mass of a collection of objects than to count them straight (using Polya counting theory). For example, the total mass of unlabeled trees of order $n$ is $n^{n-2}/n!$, because there are $n^{n-2}$ labeled trees.


So I have two questions for which a quick answer (i.e. sooner than two weeks) would be most convenient:


1. Is "mass formula" a standard name for this concept? Is there a standard name?
2. Can someone suggest a free on-line reference, comparable to a Wikipedia page or a little longer? The class textbook doesn't have a discussion.



Seeking reference for the enumerative "mass formula" concept
I do call such things "mass formulas", but then again I am a number theorist, and one of my colleagues is a quadratic form theorist who specializes in such things. So this is mostly an expression of my specific mathematical culture. 


I do not think that it is a standard term, at least not the only standard term. For instance, from another MO answer I noticed that some categorists call this the [groupoid cardinality](http://ncatlab.org/nlab/show/groupoid%20cardinality). This term in fact seems quite sensible to me, because the concept seems closely related to taking a quotient by the action of a group with nontrivial stabilizers and regarding the quotient set as a groupoid rather than a mere set.


As you say, combinatorially minded people speak of "Polya theory" or "counting with symmetry". Many algebraic geometers, upon seeing this phenomenon, would use the word "stacky". I wouldn't be surprised if there were other terms as well.


Overall I think this has the effect that a lot of people are partially rediscovering what is essentially the same concept. I would very much like to see a reasonably authoritative treatment of this subject appealing to mathematicians from different fields. Of course, I also look forward to seeing (better!) answers to this question.


