Domino tilings with barriers (Q1304629)

An Aztec diamond of order \(n\) is a region composed by \(2n(n+1)\) unit squares, arranged in bilaterally symmetric fashion as a stack of \(2n\) rows of squares, the rows having lengths \(2,4,6,\dots, 2n-2\), \(2n\), \(2n\), \(2n-2,\dots,6,4,2\). \textit{N. Elkies}, \textit{G. Kuperberg}, \textit{M. Larsen} and \textit{J. Propp} [J. Algebr. Comb. 1, No. 2, 111-132 (1992; Zbl 0779.05009)] showed that an Aztec diamond of order \(n\) can be tiled by dominoes in exactly \(2^{n(n+ 1)/2}\) ways. Here the authors pay particular attention to the line that separates the NW and the SE half of the Aztec diamond (the spine) or rather the \(2k= 2\left\lceil{n\over 2}\right\rceil\) squares that cover it. In any tiling, \(k\) of these squares belong to tiles that lie mainly in the NW half, the other \(k\) belong to tiles that lie mainly in the SE half. Prescribing for each of the \(k\) even-numbered squares whether it should belong to a tile that belongs mainly to the NW or to the SE (putting barriers toward the SE, or to the NW, resp.) is compatible with \(2^{n(n+ 1)/2}/2^k\) tilings, irrespective of the prescription. The problem can be restated in terms of determinants, and is solved using the Jakobi-Trudi identity.