A Reciprocity Theorem for Domino Tilings

Let T(m,n) denote the number of ways to tile an m-by-n rectangle with dominos. For any fixedm, the numbersT(m,n) satisfy a linear recurrence relation, and so may be extrapolated to negative values of n; these extrapolated values satisfy the relation T(m,−2−n) = εm,nT(m,n), whereεm,n = −1 if m≡ 2 (mod 4) andn is odd and whereεm,n = +1 otherwise. This is… (More)