The combinatorics of the leading root of the partial theta function

Abstract

Recently Alan Sokal studied the leading root $x_0(q)$ of the partial theta function ${\Theta }_0(x,q)=\sum _{n=0}^{\infty }{x}^n{q}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}$, considered as a formal power series. He proved that all the coefficients of $$-x_0(q)=1+q+2q^2+4q^3+9q^4+...$$ are positive integers. I give here an explicit combinatorial interpretation of these coefficients. More precisely, I show that $-x_0(q)$ enumerates rooted trees that are enriched by certain polyominoes, weighted according to their total area.

15 pages, 7 figures

Cited by in corpus (1)

• On Andrews--Warnaar's identities of partial theta functions