  We present an algorithm for evaluating a linear ``intersection transform'' of
a function defined on the lattice of subsets of an $n$-element set. In
particular, the algorithm constructs an arithmetic circuit for evaluating the
transform in ``down-closure time'' relative to the support of the function and
the evaluation domain. As an application, we develop an algorithm that, given
as input a digraph with $n$ vertices and bounded integer weights at the edges,
counts paths by weight and given length $0\leq\ell\leq n-1$ in time
$O^*(\exp(n\cdot H(\ell/(2n))))$, where $H(p)=-p\log p-(1-p)\log(1-p)$, and the
notation $O^*(\cdot)$ suppresses a factor polynomial in $n$.
