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I have tried to find the derivation of the formula for the number of tilings of an $m \times n$ board with $2 \times 1$ tiles which is the following.

$$\prod_{k=1}^{m}\prod_{l=1}^{n} \left(4\cos^2{\frac{k\pi}{m + 1} + 4\cos^2{\frac{l\pi}{n + 1}}}\right)^{\frac{1}{2}}$$

Could anyone show me how to derive it or where I could find the derivation for this incredible formula?

Tom Finet
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  • See A Course in Enumeration by Martin Aigner (Springer). – Angina Seng Dec 14 '17 at 19:57
  • You may find related informations in (https://math.stackexchange.com/questions/36099/transformations-of-domino-tilings). – Jean Marie Dec 14 '17 at 20:34
  • I just found a paper where this result first proved by Kasteleyn (1961) is established in a different way : Volker Strehl "Counting Domino Tilings of Rectangles via Resultants" Advances in Applied Mathematics 27, 597–626 (2001). You can have access to this paper on the web. – Jean Marie Dec 14 '17 at 20:40
  • Condensed version : (http://algo.inria.fr/seminars/sem01-02/strehl.ps) – Jean Marie Dec 14 '17 at 20:53
  • And here is an exposition of Kasteleyn's solution (which is quite accessible) http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1073&context=hmc_theses – Herman Tulleken Dec 15 '17 at 14:48

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