# Seeking reference for the enumerative “mass formula” concept

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 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.
• Here is another remark: The "mass formula" is a term in the Burnside counting theorem, the term corresponding to the identity permutation. Maybe this remark points to another name for the quantity? – Greg Kuperberg May 15 '10 at 5:36

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. 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.

• The name "mass formula" is standard, quite old, and not very good. The name "groupoid cardinality" is much better, but only appeared recently in the literature. (It looks like it is due to Baez and Dolan.) If there is another established name, I couldn't find it. Also the nlab page on this, and the Wikipedia page on Burnside's formula, seem to be the current documentation for the idea. So I suppose that this answer should be accepted as fair. – Greg Kuperberg Jun 16 '10 at 23:53

This doesn't qualify as a free reference, but "Graphs on surfaces and their applications" by Lando and Zvonkin has some nice examples. On p.46, after stating a theorem enumerating trees with a given "passport", the authors remark:

We will often encounter enumerative formulas where the objects are not counted one by one but a weight us assigned ti each object, and this weight is equal to 1/|Aut|, where the denominator means the order of the automorphism group of the object. Formulas of this kind are often called mass-formulas. (Footnote: The first mass-formula was proposed by H.J.S. Smith in 1867. Mass-formulas are also called Siegel–Minkowski formulas.)

Still another formulation: I recall hearing the whole idea being referred by a metaphor of skeleton and flesh. The "mass" of your example would be the "skeleton weight" or "bone mass" of the collection, the amount of "flesh" around each "bone" (the radius of the muscle ?) being the size of the automorphism group.

I find the groupoid cardinality sensible too and perfectly compatible with the basic results of Polya theory, and will certainly use it in the future.

I note that this is the converse operation of counting things with multiplicities such as roots.