School of Mathematical Sciences

On the combinatorics of the leading root of the partial theta function menu

On the combinatorics of the leading root of the partial theta function

Speaker: 
Thomas Prellberg (QMUL)
Date/Time: 
Mon, 08/10/2012 - 17:30
Room: 
M103
Seminar series: 

Recently Alan Sokal studied the leading root $x_0(q)$ of the partial theta function
$\Theta_0(x,q)=\sum\limits_{n=0}^\infty x^nq^{\binom n2}$, considered as a formal 
power series. He proved that all the coefficients of
$-x_0(q)=1+q+2q^2+4q^3+9q^4+\ldots$
are positive integers.

In this talk I will present an explicit combinatorial interpretation of these coefficients. 
More precisely, I will show that $-x_0(q)$ enumerates rooted trees that are enriched by 
certain polyominoes, weighted according to their total area.