  We extend the knowledge about so-called structural restrictions of
$\mathrm{\#SAT}$ by giving a polynomial time algorithm for $\beta$-acyclic
$\mathrm{\#SAT}$. In contrast to previous algorithms in the area, our algorithm
does not proceed by dynamic programming but works along an elimination order,
solving a weighted version of constraint satisfaction. Moreover, we give
evidence that this deviation from more standard algorithm is not a coincidence,
but that there is likely no dynamic programming algorithm of the usual style
for $\beta$-acyclic $\mathrm{\#SAT}$.
