Combinatorial Algorithms

Volume 7056 of the series Lecture Notes in Computer Science pp 223-235

Enumerating Tatami Mat Arrangements of Square Grids

  • Alejandro EricksonAffiliated withDepartment of Computer Science, University of Victoria
  • , Mark SchurchAffiliated withMathematics and Statistics, University of Victoria

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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.