TAILIEUCHUNG - Báo cáo toán học: "Monomer-dimer tatami tilings of rectangular regions"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Monomer-dimer tatami tilings of rectangular regions. | Monomer-dimer tatami tilings of rectangular regions Alejandro Erickson Frank Ruskey Jennifer Woodcock ate@ ruskey@ jwoodcoc@ Department of Computer Science University of Victoria PO Box 3055 STN CSC Victoria BC V8W 3P6 Canada Mark Schurch mschurch@ Department of Mathematics and Statistics University of Victoria PO Box 3060 STN CSC Victoria BC V8W 3R4 Canada Submitted Mar 4 2011 Accepted May 3 2011 Published May 16 2011 Mathematics Subject Classification 52C20 05B45 05A19 Abstract In this paper we consider tilings of rectangular regions with two types of tiles 1 X 2 tiles dimers and 1 X 1 tiles monomers . The tiles must cover the region and satisfy the constraint that no four corners of the tiles meet such tilings are called tatami tilings. We provide a structural characterization and use it to prove that the tiling is completely determined by the tiles that are on its border. We prove that the number of tatami tilings of an n X n square with n monomers is n2n-1. We also show that for fixed-height the generating function for the number of tatami tilings of a rectangle is a rational function and outline an algorithm that produces the generating function. Keywords tatami monomer-dimer tiling rational generating function 1 What is a tatami tiling Traditionally a tatami mat is made from a rice straw core with a covering of woven soft rush straw. Originally intended for nobility in Japan they are now available in mass-market stores. The typical tatami mat occurs in a 1 X 2 aspect ratio and various configurations of them are used to cover floors in houses and temples. By parity considerations it may be necessary to introduce mats with a 1 X 1 aspect ratio in order to cover the floor of a room. Such a covering is said to be auspicious if no four corners of mats THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P109 1 Figure 1 a Vertical bond pattern. b Horizontal bond pattern. c Herringbone pattern. Figure 2 What is the least number of monomers .

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