Enumerating Tatami Mat Arrangements of Square Grids
We prove that the number of monomer-dimer tilings of an n×n square grid, with m < n monomers in which no four tiles meet at any point is m2 m + (m + 1)2 m + 1, when m and n have the same parity. In addition, we present a new proof of the result that there are n2 n − 1 such tilings with n monomers, which divides the tilings into n classes of size 2 n − 1. The sum of these over all m ≤ n has the closed form 2 n − 1(3n − 4) + 2 and, curiously, this is equal to the sum of the squares of all parts in all compositions of n.
KeywordsRice Straw Discrete Apply Mathematic Clockwise Vortex Dime Covering Regular Grammar
Unable to display preview. Download preview PDF.
- 1.Alhazov, A., Morita, K., Iwamoto, C.: A note on tatami tilings. In: Proceedings of the 2009 LA Winter Symposium Mathematical Foundation of Algorithms and Computer Science, vol. 1691, pp. 1–7 (2010)Google Scholar
- 6.Jovovic, V.: Comment on a027992 (2005), http://oeis.org/A027992
- 7.Knuth, D.E.: The Art of Computer Programming, vol 4A: Combinatorial Algorithms, Part 1, 1st edn. Addison-Wesley Professional (2011)Google Scholar
- 9.Morrison, P., Morrison, P.: 100 or so books that shaped a century of science. American Scientist 87(6), 1 (1999)Google Scholar
- 10.Piesk, T.: Binary and compositions 5 (2010), http://commons.wikimedia.org/wiki/File:Binary_and_compositions_5.svg