Domino Tilings with Barriers

In this paper, we continue the study of domino-tilings of Aztec diamonds. In particular, we look at certain ways of placing “barriers” in the Aztec diamond, with the constraint that no domino may cross a barrier. Remarkably, the number of constrained tilings is independent of the placement of the barriers. We do not know of a simple combinatorial explanation of this fact; our proof uses the Jacobi–Trudi identity.