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Cell Growth Problems

  • David A. Klarner (a1)
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The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.

The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn , which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn , or (ii) in Tn , if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.

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References
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1. Eden, M., A two-dimensional growth process, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. IV (Berkeley, California, 1961), pp. 223239.
2. Golomb, S. W., Checkerboards and polyominoes, Amer. Math. Monthly, 61 (1954), 275282.
3. Harary, F., Unsolved problems in the enumeration of graphs, (Magyar Tud. Akad. Mat. Kutato Int Kozl.) Publ. Math. Inst. Hungar. Acad. Sci., 5 (1960), 6395.
4. Harary, F., “Combinatorial problems in graphical enumeration,” Chap. 6, Applied combinatorial analysis, ed. by Beckenbach, F. (New York, 1964), pp. 185217.
5. Klarner, D. A., Some results concerning polyominoes, Fibonacci Quarterly 8 (1965), 920.
6. Klarner, D. A., Combinatorial problems involving the Fredholm integral equation, to appear.
7. Pólya, G. and Szego, G., Aufgaben und Lehrsätze aus der Analysis, Vol. 1 (Berlin, 1925).
8. Read, R. C., Contributions to the cell growth problem, Can. J. Math., 14 (1962), 120.
9. Riesz, F. and Sz-Nagy, B., Functional analysis (New York, 1955).
10. Titchmarsh, E. C., The theory of functions, 2nd ed. (Oxford, 1939).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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