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Journal of Mathematical Sciences

, Volume 200, Issue 6, pp 647–653 | Cite as

Domino Tilings and Determinants

  • V. Aksenov
  • K. Kokhas
Article

Consider an arbitrary simply connected figure F on the square grid and its dual graph (vertices correspond to cells, edges correspond to cells sharing a common side). We investigate the relationship between the determinant of the adjacency matrix of the graph and the domino tilings of the figure F. We prove that in the case where all the tilings can be split into pairs such that the numbers of vertical dominoes in each pair differ by one, then det A F = 0. And in the case where all the tilings except one can be split into such pairs, det A F = (−1) s , where s is half the area of the figure F. Bibliography: 6 titles.

Keywords

Adjacency Matrix Integer Point Dual Graph Black Vertex White Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    M. N. Vyalyi, “Pfaffians, or the art to attach signs…,” Mat. Prosveschenie, Ser. 3, No. 9, 129–142 (2005).Google Scholar
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    D. V. Karpov, “On the parity of the number of domino tilings,” unpublished (1997).Google Scholar
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    K. P. Kokhas, “Domino tilings,” Mat. Prosveschenie, Ser. 3, No. 9, 143–163 (2005).Google Scholar
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    K. P. Kokhas, “Domino tilings of aztec diamonds and squares,” Zap. Nauchn. Semin. POMI, 360, 180–230 (2008).Google Scholar
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    D. Zeilberger, “A combinatorial approach to matrix algebra,” Discrete Math., 56, 61–72 (1985).CrossRefzbMATHMathSciNetGoogle Scholar
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    D. Pragel, “Determinants of box products of paths,” http://arxiv.org/1110.3497.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.St.Petersburg National Research University of Information Technologies, Mechanics, and OpticsSt. PetersburgRussia
  2. 2.St.Petersburg State UniversitySt. PetersburgRussia

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