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Domino tilings of Aztec diamonds and squares

An Aztec diamond of rank n is a rhombus of side length n on the square grid. We give several new combinatorial proofs of known theorems about the numbers of domino tilings of diamonds and squares. We also prove generalizations of these theorems for the generating polynomials of some statistics of tilings. Some results here are new. For example, we describe how to calculate the rank of a tiling using special weights of edges on the square grid. Bibliography: 17 titles.

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Correspondence to K. P. Kokhas.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 180–230.

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Kokhas, K.P. Domino tilings of Aztec diamonds and squares. J Math Sci 158, 868–894 (2009). https://doi.org/10.1007/s10958-009-9421-3

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Keywords

  • Russia
  • Side Length
  • Special Weight
  • Aztec
  • Combinatorial Proof