Domino Tilings of Shifted Two-Block Regions: Exact Identities, Equal-Height Theorems, and Computational Evidence

We study domino tilings of a rectangle and of a natural family of shifted two-block regions. A seam decomposition gives an exact identity for the difference between the rectangular tiling count and the shifted count. We prove, in particular, that every nontrivially shifted equal-height two-block region is strictly beaten by the corresponding rectangle. We also prove a same-sign result for the case in which one block has height one and the other has odd height, and derive an exact reflection-based bilinear identity for general heights. Several finite computations and machine-checked finite or parametric components are recorded with their scopes kept separate. The case of general unequal heights remains open. In particular, the q = 6 all-root decomposition presented here is an exact candidate certificate only, not a theorem: the coefficient/decomposition implication needed to prove positivity has not been formalized, and the Rectangle-Optimal conjecture is not proved.