Counting Paths and Packings in Halves

We show that one can count k-edge paths in an n-vertex graph and m-set k-packings on an n-element universe, respectively, in time \({n \choose k/2}\) and \({n \choose mk/2}\), up to a factor polynomial in n, k, and m; in polynomial space, the bounds hold if multiplied by 3 k/2 or 5 mk/2, respectively. These are implications of a more general result: given two set families on an n-element universe, one can count the disjoint pairs of sets in the Cartesian product of the two families with O(n l) basic operations, where l is the number of members in the two families and their subsets.