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 ST-CONNECTIVITY 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：我们证明了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莫比乌斯函数值。利用这一结论，我们确定了无环箭图（有向无环多重图）中ST-连通性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，其系数为$(-1)^D$，其中$D$是箭图（对于平面图而言为平面区域数）的易计算函数，且该箭图与单项式对应。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$2^{\Omega(n^2)}$。