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We show that #SAT on β-acyclic CNF-formulas can be solved in polynomial time. In contrast to previous algorithms for other structurally restricted classes of formulas, our algorithm does not proceed by dynamic programming. Instead, it works along an elimination order, solving a weighted version of constraint satisfaction. We give evidence that this deviation from more standard algorithms is no coincidence by showing that it is outside of the framework recently proposed by Saether et al. (SATfatcat:2whjx75fgnerbealyulws2pxhu