  In this paper we describe the Hopf algebra approach to the center problem for
the differential equation $\frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x)v^{i+1}$,
$x\in [0,T]$, and study some combinatorial properties of the first return map
of this equation. The paper summarizes and extends previously developed
approaches to the center problem due to Devlin and the author.
