Abstract
Polyominoes are edge-connected sets of cells on the square lattice ℤ2. We investigate polyominoes on a square lattice embedded on so-called twisted cylinders of a bounded width (perimeter) w. We prove that the limit growth rate of polyominoes of the latter type approaches that of polyominoes of the former type, as w tends to infinity. We also prove that for any fixed value of w, the formula enumerating polyominoes on a twisted cylinder of width w satisfies a linear recurrence whose complexity grows exponentially with w. By building the finite automaton that “grows” polyominoes on the twisted cylinder, we obtain the prefix of the sequence enumerating these polyominoes. Then, we recover the recurrence formula by using the Berlekamp-Massey algorithm.
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References
Barequet, G.: Recurrence data for polyominoes on twisted cylinders, http://www.cs.technion.ac.il/~barequet/twisted/
Barequet, G., Moffie, M.: On the complexity of Jensen’s algorithm for counting fixed polyominoes. J. Discrete Algorithms 5, 348–355 (2007)
Barequet, G., Moffie, M., Ribó, A., Rote, G.: Counting polyominoes on twisted cylinders. Integers 6, #A22 (2006)
Brent, R.P., Gustavson, F.G., Yun, D.Y.Y.: Fast solution of Toeplitz systems of equations and computation of Padé approximant. J. Algorithms 1, 259–295 (1980)
Gaunt, D.S., Sykes, M.F., Ruskin, H.: Percolation processes in d-dimensions. J. Phys. A—Math. Gen. 9, 1899–1911 (1976)
Jensen, I.: Enumerations of lattice animals and trees. J. Stat. Phys. 102, 865–881 (2001)
Jensen, I.: Counting polyominoes: A parallel implementation for cluster computing. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J., Zomaya, A.Y. (eds.) ICCS 2003, Part III. LNCS, vol. 2659, pp. 203–212. Springer, Heidelberg (2003)
Klarner, D.A.: Cell growth problems. Can. J. Math. 19, 851–863 (1967)
Klarner, D.A., Rivest, R.L.: A procedure for improving the upper bound for the number of n-ominoes. Can. J. Math. 25, 585–602 (1973)
Madras, N.: A pattern theorem for lattice clusters. Ann. Comb. 3, 357–384 (1999)
Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE T. Inf. Theory 15, 122–127 (1969)
Reeds, J.A., Sloane, N.J.A.: Shift-register synthesis (modulo m). SIAM J. Comput. 14, 505–513 (1985)
Stanley, R.: Enumerative Combinatorics, I. Cambridge University Press (1997)
Wynn, P.: On a device for computing the e m (S n ) transformation. Math. Tables and other Aids to Computation 10, 91–96 (1956)
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Aleksandrowicz, G., Asinowski, A., Barequet, G., Barequet, R. (2014). Formulae for Polyominoes on Twisted Cylinders. In: Dediu, AH., Martín-Vide, C., Sierra-Rodríguez, JL., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2014. Lecture Notes in Computer Science, vol 8370. Springer, Cham. https://doi.org/10.1007/978-3-319-04921-2_6
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DOI: https://doi.org/10.1007/978-3-319-04921-2_6
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