We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a \stc\ problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{\Omega(n^2)}$.

翻译：我们证明，任意单调布尔函数的表示多项式的系数是与该函数相关的原子格上的默比乌斯函数值。利用这一结论，我们确定了无环箭图（有向无环多重图）中 \stc\ 问题对应的任意布尔函数的表示多项式。只有对应于路径并集的单项式具有非零系数，其值为 $(-1)^D$，其中 $D$ 是关于该单项式对应箭图的一个易于计算的函数（对于平面图，$D$ 即为平面区域的数量）。我们进一步确定，对于二维 $n \times n$ 网格连通性问题，具有非零系数的单项式数量为 $2^{\Omega(n^2)}$。