Oops, my claim that "a torus spanning configuration is necessarily a spanning configuration" is false. Some torus spanning configurations that loop more than once may not be spanning configurations. Tom Tom Karzes writes:
Variants of possible interest:
1. "Double-spanning" configurations which have both horzontal and vertical paths
2. Torus spanning configurations, which have a "north-south" loop on a torus. Note that a torus spanning configuration is necessarily a spanning configuration, but the reverse is not always true. Note that some paths may loop multiple times before connecting back.
3. Torus double-spanning configurations, which contain loops along both axes.
Tom
Cris Moore via math-fun writes:
My friend Stephan Mertens and I just proved something you might enjoy. Consider filling a subset of the cells of an n-by-m rectangular lattice. Call the resulting configuration “spanning" if there is a path of occupied cells from the top row to the bottom row (stepping orthogonally).
Let N_even (N_odd) be the number of spanning configurations with an even (resp. odd) number of occupied cells. Show that N_even-N_odd = +1 or -1, and determine the sign as a function of n and m. As a corollary, the total number of spanning configurations is odd.
For instance, for n=m=2 there are 7 spanning configurations, with N_even=3 and N_odd=4.
Cris
Cris Moore moore@santafe.edu