Abstract
We introduce tatami tilings, and present some of the many interesting questions that arise when studying them. Roughly speaking, we are considering tilings of rectilinear regions with 1 ×2 dimer tiles and 1 ×1 monomer tiles, with the constraint that no four corners of the tiles meet. Typical problems are to minimize the number of monomers in a tiling, or to count the number of tilings in a particular shape. We determine the underlying structure of tatami tilings of rectangles and use this to prove that the number of tatami tilings of an n ×n square with n monomers is n2n − 1. We also prove that, for fixed-height, the number of tatami tilings of a rectangle is a rational function and outline an algorithm that produces the coefficients of the two polynomials of the numerator and the denominator.
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References
Kenyon, R., Okounkov, A.: What is ... a dimer? Notices of the American Mathematical Society 52, 342–343 (2005)
Stanley, R.P.: On dimer coverings of rectangles of fixed width. Discrete Applied Mathematics 12(1), 81–87 (1985)
Knuth, D.E.: The Art of Computer Programming, vol. 4, fascicle 1B. Addison-Wesley, Reading (2009)
Ruskey, F., Woodcock, J.: Counting fixed-height tatami tilings. Electronic Journal of Combinatorics 16, 20 (2009)
Dürr, C., Guiñez, F., Matamala, M.: Reconstructing 3-colored grids from horizontal and vertical projections is NP-hard. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 776–787. Springer, Heidelberg (2009)
Anzalone, N., Baldwin, J., Bronshtein, I., Petersen, T.K.: A reciprocity theorem for monomer-dimer coverings. In: Morvan, M., Rémila, É. (eds.) Discrete Models for Complex Systems, DMCS ’03. DMTCS Proceedings, Discrete Mathematics and Theoretical Computer Science, vol. AB, pp. 179–194 (2003)
Pachter, L.: Combinatorial approaches and conjectures for 2-divisibility problems concerning domino tilings of polyominoes. Electronic Journal of Combinatorics 4, 2–9 (1997)
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Erickson, A., Ruskey, F., Schurch, M., Woodcock, J. (2010). Auspicious Tatami Mat Arrangements. In: Thai, M.T., Sahni, S. (eds) Computing and Combinatorics. COCOON 2010. Lecture Notes in Computer Science, vol 6196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14031-0_32
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DOI: https://doi.org/10.1007/978-3-642-14031-0_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14030-3
Online ISBN: 978-3-642-14031-0
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