  In this paper, we build on the idea of Valiant \cite{Val79a} and
Ben-Dor/Halevi \cite{Ben93}, that is, to count the number of satisfying
solutions of a boolean formula via computing the permanent of a specially
constructed matrix. We show that the Desnanot-Jacobi identity ($\dji$) prevents
Valiant's original approach to achieve a parsimonious reduction to the
permanent over a field of characteristic two. As the next step, since the
computation of the permanent is $#\classP$-complete, we make use of the
equality of the permanent and the number of perfect matchings in an unweighted
graph's bipartite double cover. Whenever this bipartite double cover (BDC) is
planar, the number of perfect matchings can be counted in polynomial time using
Kasteleyn's algorithm \cite{Kas67}. To enforce planarity of the BDC, we replace
Valiant's original gadgets with new gadgets and describe what properties these
gadgets must have. We show that the property of \textit{circular planarity}
plays a crucial role to find the correct gadgets for a counting problem. To
circumvent the $\dji$-barrier, we switch over to fields
$\mathbb{Z}/p\mathbb{Z}$, for a prime $p > 2$.
  With this approach we are able to count the number of solutions for
$\forestdreisat$ formulas in randomized polynomial time. Finally, we present a
conjecture that states which kind of generalized gadgets can not be found,
since otherwise one could prove $\classRP = \classNP$. The conjecture
establishes a relationship between the determinants of the minors of a graph
$\grG$'s adjacency matrix and the \textit{circular planar} structure of
$\grG$'s BDC regarding a given set of nodes.
