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Polyhex


Polyhexes

Polyhex 是 polyominoes (多米诺骨牌) 和 polyiamonds (多钻石形) 的类似物,其中正六边形的集合以相邻边排列。它们也称为 hexes、hexas 或 polyfrobs (Beeler 1972)。对于 4-hexes (tetrahexes),可能的排列方式被称为 bee (蜜蜂形), bar (条形), pistol (手枪形), propeller (螺旋桨形), worm (蠕虫形), arch (拱形), 和 wave (波浪形)。

几何平面 n-polyhexes 的数量,对于 n=1, 2, ... 分别是 1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, ... (OEIS A000228; Klarner 1967, Balaban and Harary 1968, Harary and Read 1970, Lunnon 1972, Gardner 1978, Knop et al. 1984, Gardner 1988),

PolyhexesWithHoles

带有孔洞的 n-polyhexes 的数量,对于 n=6, 7, 8, ... 分别是 1, 2, 13, 67, 404, ... (OEIS A038144; Myers),其中前几个示例如上所示。

PolyhexesOneSided

“单侧” polyhexes 被认为是固定在平面上的,因此镜像图像被单独计数。n-六边形单侧 polyhexes 的数量分别是 1, 1, 3, 10, 33, 147, 620, 2821, 12942, 60639, 286190, 1364621, 6545430, ... (OEIS A006535)。

一个简单的连通 polyhex 被称为 fusene (并六边形)。


另请参阅

Benzenoid, Fullerene, Fusene, Polyform, Polyhex Tiling, Polyiamond, Polyomino, Polyplet

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参考文献

Balaban, A. T. "Enumeration of Cyclic Graphs." 在 Chemical Applications of Graph Theory (编 A. T. Balaban). London: Academic Press, 页 63-105, 1976.Balaban, A. T. and Harary, F. "Chemical Graphs V: Enumeration and Proposed Nomenclature of Benzenoid Cata-Condensed Polycyclic Aromatic Hydrocarbons." Tetrahedron 24, 2505-2506, 1968.Balasubramanian, K.; Kauffman, J. J.; Koski, W. S.; and Balaban, A. T. "Graph Theoretical Characterization and Computer Generation of Certain Carcinogenic Benzenoid Hydrocarbons and Identification." J. Comput. Chem. 1, 149-157, 1980.Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, 页 48-50, 1972年2月. http://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item112.Beineke, L. W. and Pippert, R. E. "On the Enumeration of Planar Trees of Hexagons." Glasgow Math. J. 15, 131-147, 1974.Brinkmann, G.; Caporossi, G.; and Hansen, P. "A Constructive Enumeration of Fusenes and Benzenoids." J. Algorithms. 45, 155-166, 2002.Brinkmann, G.; Caporossi, G.; and Hansen, P. "A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons." J. Chem. Inf. Comput. Sci. 43, 842-851, 2003.Clarke, A. L. "Polyhexes." http://www.recmath.com/PolyPages/PolyPages/Polyhexes.html.Cyvin, S. J.; Brunvoll, J.; Xiaofeng, G.; and Fuji, Z. "Number of Perifusenes with One Internal Vertex." Rev. Roumaine Chem. 38, 65-77, 1993.Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydrocarbons. 1. Isomer Enumeration of Fused Polycyclic Aromatic Hydrocarbon." J. Chem. Inf. Comput. Sci. 22, 15-22, 1982.Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydrocarbons. 2. Polycyclic Aromatic Hydrocarbons Containing Tetragonal, Pentagonal, Heptagonal, and Octagonal Rings." J. Chem. Inf. Comput. Sci. 22, 139-152, 1982.Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydrocarbons. 3. Enumeration of All the Polycyclic Conjugated Isomers of Pyrene Having Ring Sizes Ranging from 3 to 9." Math. Chem (Mülheim/Ruhr) 14, 83-138, 1983.Gardner, M. "Polyhexes and Polyaboloes." 章 11 在 Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, 页 146-159, 1978.Gardner, M. "Tiling with Polyominoes, Polyiamonds, and Polyhexes." 章 14 在 Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, 页 175-187, 1988.Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 第二版. Princeton, NJ: Princeton University Press, 页 92-93, 1994.Harary, F. "Graphical Enumeration Problems." 在 Graph Theory and Theoretical Physics (编 F. Harary). London: Academic Press, 页 1-41, 1967.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 页 178-197, 1994.Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973.Harary, F. and Read, R. C. "The Enumeration of Tree-Like Polyhexes." Proc. Edinburgh Math. Soc. 17, 1-13, 1970.Keller, M. "Counting Polyforms." http://members.aol.com/wgreview/polyenum.html.Klarner, D. A. "Cell Growth Problems." 在 Canad. J. Math 19, 851-863, 1967.Knop, J. V.; Szymanski, K.; Jeričević, Ž.; and Trinajstić, N. "On the Total Number of Polyhexes." Match: Commun. Math. Chem., 第 16号, 119-134, 1984年8月.Lunnon, W. F. "Counting Hexagonal and Triangular Polyominoes." 在 Graph Theory and Computing (编 R. C. Read). New York: Academic Press, 页 87-100, 1972.Myers, J. "Polyomino Tiling." http://www.srcf.ucam.org/~jsm28/tiling/.Palmer, E. M. "Variations of the Cell Growth Problem." 在 Graph Theory and Applications: Proceedings of the Conference at Western Michigan University, Kalamazoo, Mich., May 10-13, 1972 (编 Y. Alavi, D. R. Lick, and A. T. White). New York: Springer-Verlag, 页 214-223, 1972.Sloane, N. J. A. 序列 A000228/M2682, A038144, 和 A006535/M2846 在“整数序列在线百科全书”中。Vichera, M. "Polyforms." http://www.vicher.cz/puzzle/polyforms.htm.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, 页 342-343, 1993.Weisstein, E. W. "Books about Polyominoes." http://www.ericweisstein.com/encyclopedias/books/Polyominoes.html.

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Polyhex

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Weisstein, Eric W. "Polyhex." 来自 MathWorld——Wolfram Web 资源。 https://mathworld.net.cn/Polyhex.html

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