Abstract
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 m2m + (m + 1)2m + 1, when m and n have the same parity. In addition, we present a new proof of the result that there are n2n − 1 such tilings with n monomers, which divides the tilings into n classes of size 2n − 1. The sum of these over all m ≤ n has the closed form 2n − 1(3n − 4) + 2 and, curiously, this is equal to the sum of the squares of all parts in all compositions of n.
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Erickson, A., Schurch, M. (2011). Enumerating Tatami Mat Arrangements of Square Grids. In: Iliopoulos, C.S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2011. Lecture Notes in Computer Science, vol 7056. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25011-8_18
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DOI: https://doi.org/10.1007/978-3-642-25011-8_18
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