User federico ardila - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T08:13:50Z http://mathoverflow.net/feeds/user/21688 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101191/generating-functions-tutte-polynomials-and-the-bivariate-series-sum-n-xn-y Generating functions, Tutte polynomials, and the bivariate series $\sum_n x^n y^{n^2} / n!$. Federico Ardila 2012-07-03T00:48:34Z 2012-07-03T17:56:50Z <p>A few years ago I <a href="http://math.sfsu.edu/federico/Articles/arrangem.pdf" rel="nofollow">computed</a> 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? </p> http://mathoverflow.net/questions/86066/how-many-tutte-polynomials-of-complete-graphs-are-known/101193#101193 Answer by Federico Ardila for How many Tutte polynomials of complete graphs are known? Federico Ardila 2012-07-03T01:03:20Z 2012-07-03T01:03:20Z <p>There is a simple formula for the generating function of $T_{K_n}(x,y)$, which is more cleanly expressed in terms of the (equivalent) "coboundary polynomial" $X_M(q,t) = (t-1)^{r(M)} T_M(1+\frac{q}{t-1}, t)$: $$1+q\sum_{n \geq 1} X_{K_n}(q,t) \frac{x^n}{n!} = \left( \sum_{n \geq 0} t^{n \choose 2}\frac{x^n}{n!}\right)^q.$$ This is essentially due to Tutte in the 50s; see <a href="http://math.sfsu.edu/federico/Articles/arrangem.pdf" rel="nofollow">this paper</a> and the references in it.</p> <p>For what it's worth, using similar methods one easily obtains formulas for other similar families such as <a href="http://www.math.umn.edu/~reiner/Papers/cyclotomic.pdf" rel="nofollow">complete bipartite graphs</a> (for graphs) and <a href="http://math.sfsu.edu/federico/Articles/arrangem.pdf" rel="nofollow">classical root systems</a> (for matroids).</p> http://mathoverflow.net/questions/101191/generating-functions-tutte-polynomials-and-the-bivariate-series-sum-n-xn-y Comment by Federico Ardila Federico Ardila 2012-07-03T17:58:18Z 2012-07-03T17:58:18Z Ira, I did mean that, and I corrected it in the post. (I hope that's proper math overflow etiquette.)