Cris, That’s a really fun fact! (I have no idea how to prove it as of yet.) Are there any patterns mod 4 for the total number of spanning configurations? Jim On Mon, Sep 2, 2019 at 4:37 PM Cris Moore via math-fun < math-fun@mailman.xmission.com> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun