  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 $2^{\Omega(n^2)}$.
