ar X iv : m at h / 98 01 06 7 v 2 [ m at h . C O ] 6 J an 1 99 9 Domino tilings with barriers

In this paper, we continue the study of domino-tilings of Aztec diamonds (introduced in [1] and [2]). 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. (NOTE: This article has been published in the Journal of Combinatorial Theory, Series A, the only definitive repository of the content that has been certified and accepted after peer review. An Aztec diamond of order n is a region composed of 2n(n+1) unit squares, arranged in bilaterally symmetric fashion as a stack of 2n rows of squares, the rows having lengths A domino is a 1-by-2 (or 2-by-1) rectangle. It was shown in [1] that the Aztec diamond of order n can be tiled by dominoes in exactly 2 n(n+1)/2 ways. Here we study barriers, indicated by darkened edges of the square grid associated with an Aztec diamond. These are edges that no domino is permitted to cross. (If one prefers to think of a domino tiling of a region as a perfect matching of a dual graph whose vertices correspond to grid-squares and whose edges correspond to pairs of grid-squares having a shared edge, then putting down a barrier in the tiling is tantamount to removing an edge from the dual graph.) Figure 1(a) shows an Aztec diamond of order 8 with barriers, and Figure 1(b) shows a domino-tiling that is compatible with this placement of barriers. The barrier-configuration of Figure 1(a) has special structure. Imagine a line of slope 1 running through the center of the Aztec diamond (the " spine "), passing through 2k grid-squares, with k = ⌈n/2⌉. Number these squares from lower left (or " southwest ") to upper right (or " northeast ") as squares 1 through 2k. For each such square, we may place barriers on its bottom and right edges (a " zig "), barriers on its left and top edges (a " zag "), or no barriers at all (" zip "). Thus Figure 1 corresponds to the sequence of decisions " zip, zig, zip, zag, zip, zag, zip, zig. " Notice that in this example, for all i, the ith square has a zig or …