A few years ago I computed the Tutte polynomials of the matroids given by the classical Coxeter groups, and found that their generating functions are all simple variations of the series $\sum_n \frac{x^n y^{n^2}}{n!}$. I've wondered if there is a more geometric/algebraic explanation of this. Is this series known? Are there other natural occurrences of it that might be relevant?

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    $\begingroup$ en.wikipedia.org/wiki/Theta_function $\endgroup$ – Qiaochu Yuan Jul 3 '12 at 0:58
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    $\begingroup$ Don't you mean $\sum_n x^n y^{n^2}/n!$? $\endgroup$ – Ira Gessel Jul 3 '12 at 13:39
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    $\begingroup$ The generating function $f(x) = \sum_n x^n y^\binom{n}{2}/n!$ satisfies $f'(x) = f(xy)$. $\endgroup$ – Ira Gessel Jul 3 '12 at 17:17
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    $\begingroup$ So writing $g(u)=f(\exp(u))$, Ira's equation becomes a "delay differential equation" where the derivative $g'(u)$ is written in terms of $g(u-\tau)$, for some $\tau$. $\endgroup$ – Gerald Edgar Jul 3 '12 at 18:49
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    $\begingroup$ I am not sure if this might lead to anything but $$ \sum_{n\geqslant0}\frac{e^{nz}e^{n^2t}}{n!}=e\sum_{k,l\geqslant0} \frac{{B_{k+2l}}}{{k!l!(k+l)!}}z^kt^l $$ where the $B$'s are the Bell numbers... $\endgroup$ – მამუკა ჯიბლაძე Apr 8 '14 at 7:24

Here is an attempt at a "soft" answer inspired by a paper that appeared after this question was asked here. The arithmetic Tutte polynomials of the classical root systems by Ardila, Castillo, and Henley. In particular this will not contain anything the OP doesn't already now :-)

There are two methods used for the computation of these (classic or arithmetic) Tutte polynomial generating functions. The finite field method by Ardila mentioned in the question has wide applicability, but in the paper linked above there is a different way to compute this generating function, which I believe give some insight on the presence of $F(x,y)=\sum_{n\geq 0} x^ny^{\binom{n}{2}}/n!$.

Using this generating function we can count one of the most fundamental combinatorial objects: graphs according to the number of connected components, number of edges, and number of vertices (the particular function is $F(x,1+y)^z$). On the other hand, through a series of operations in the sense of combinatorial species, one can obtain the generating function for some combinatorial objects called signed graphs, introduced by Zaslavsky, which serve as a combinatorial model encoding the relevant statistics of the classical root systems. Therefore one should expect the generating functions in question, to be computable from combinations of $F$ together with operations like multiplication, exponentiation, and composition.

  • $\begingroup$ Thank you very much for your answer, it certainly provokes a healthy mindshift. Can you provide any links/hints to the treatment of any "theta-like" series in terms of Joyal's species? $\endgroup$ – მამუკა ჯიბლაძე Jan 14 '15 at 6:25

I don't know if this might help, but Alan Sokal has extensively studied this kind of series (with $n$ choose 2 instead of the $n^2$ exponent for $y$). See the material for these recent lectures.


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