In article <email@example.com>, David Corfield <firstname.lastname@example.org> wrote:
>I have a few questions about combinatorics and species, and would be >grateful for any comments.
I'll tackle the easiest one now and attempt the harder ones later, but I sure hope other people try too.
>X/(1 - e^X) looks like a simple composition of species - pick out a one >element set and arrange a set of sets whose union is the remainder - yet >it can't be that simple to get at the Bernoulli numbers. I guess lots of >unwanted empty sets appear in the union.
I don't see what you're worrying about here, but I presume it's related to the naive "0/0" you get when you evaluate this expression at X= 0.
I've been meaning to think about this ever since I read Connes' comments on Bernoulli numbers in this book:
Alain Connes, Andre Lichnerowicz and Marcel Paul Schutzenberger, A Triangle of Thoughts, AMS, Providence, 2000.
He points out that if H is the Hamiltonian for some sort of particle in a box and beta is the inverse temperature,
is the operator you take the trace of to get the partition function of a collection of an arbitrary number of particles of this sort. And he claims that pondering this explains all the appearances of X/(1 - e^X) and the Bernoulli numbers in topology! See Milnor and Stasheff's book "Characteristic Classes" for an introduction to *that* - but this book was written before quantum theory invaded topology, so we're left to fit Connes' clues together for ourselves.
>Why is it that you get the series expansions for species if they don't >blow up for X= 0, yet you're most interested in X= 1?
Well, for now I'll just say that that species don't "blow up"; it's only when you decategorify them that you get divergent formal power series. Then of course they usually diverge at X = 1, because we are usually interested in structures that can be put on finite sets in infinitely many different ways.
You probably knew all this and wanted a deeper answer; I don't think there is one. I forget if you know what it means to evaluate a species at an arbitrary groupoid X; if not, maybe this would make you happier.