Domino Tilings and Determinants
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 AF = 0. And in the case where all the tilings except one can be split into such pairs, det AF = (−1)s, where s is half the area of the figure F. Bibliography: 6 titles.
Unable to display preview. Download preview PDF.
- 1.M. N. Vyalyi, “Pfaffians, or the art to attach signs…,” Mat. Prosveschenie, Ser. 3, No. 9, 129–142 (2005).Google Scholar
- 2.D. V. Karpov, “On the parity of the number of domino tilings,” unpublished (1997).Google Scholar
- 3.K. P. Kokhas, “Domino tilings,” Mat. Prosveschenie, Ser. 3, No. 9, 143–163 (2005).Google Scholar
- 4.K. P. Kokhas, “Domino tilings of aztec diamonds and squares,” Zap. Nauchn. Semin. POMI, 360, 180–230 (2008).Google Scholar
- 6.D. Pragel, “Determinants of box products of paths,” http://arxiv.org/1110.3497.