On 2-adic behavior of the number of domino tilings on torus

Authors

  • Weidong Cheng Chongqing University of Posts and Telecommunications
  • Xuejun Guo Nanjing University

DOI:

https://doi.org/10.11575/cdm.v14i1.62673

Abstract

We study the 2-adic behavior of the number of domino tilings of a $2(2n+1)\times 2(2n+1)$ torus. We show that this number is of the form $2^{4n+3}g(n)^2+2^{8n+2}(2n+1)^{4n}h(n)$, where $g(n)$ and $h(n)$ are odd positive integers. Moreover, we prove that $g(n)$ and $h(n)$ are uniformly continuous under the 2-adic metric and invariant under interchanging $n$ and $-1-n$. This paper is an analogy of Henry Cohn's results for $2n\times 2n$ squares (Electron. J. Combin. 6 (1999)).

References

[1] Henry Cohn, 2-adic behavior of numbers of domino tilings, Electron. J. Combin. 6 (1999), Research Paper 14, 7. MR1667454

[2] Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), no. 2, 297-346. MR1815214

[3] Fernando Q. Gouvea, p-adic numbers, second ed., Universitext, Springer-Verlag, Berlin, 1997, An introduction. MR1488696

[4] Nathan Jacobson, Basic algebra. I, second ed., W. H. Freeman and Company, New York, 1985. MR780184

[5] P. W. Kasteleyn, The statistics of dimers on a lattice : I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), no. 12, 1209-1225.

[6] Jurgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999, Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder. MR1697859

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Published

2019-12-25

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Articles