From Linear Partitions to Parallelogram Polyominoes

  • Roberto Mantaci
  • Paolo Massazza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)


We provide a bijection between parallelogram polyominoes and suitable pairs of linear partitions. This lets us design a CAT (Constant Amortized Time) algorithm for generating all parallelogram polyominoes of size n using \(O(\sqrt n)\) space.


Polyominoes Exhaustive generation CAT algorithms 


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  1. 1.
    Barcucci, E., Frosini, A., Rinaldi, S.: Direct-convex polyominoes: ECO method and bijective results. In: Brak, R., Foda, O., Greenhill, C., Guttman, T., Owczarek, A. (eds.) Proceedings of Formal Power Series and Algebraic Combinatorics 2002, Melbourne (2002)Google Scholar
  2. 2.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from horizontal and vertical projections. Theoret. Comp. Sci. 155(2), 321–347 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barcucci, E., Del Lungo, A., Pergola, E., Pinzani, R.: ECO:a methodology for the enumeration of combinatorial objects. J. of Diff. Eq. and App. 5, 435–490 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bousquet-Mélou, M.: A method for the enumeration of various classes of column-convex polygons. Discrete Math. 154, 1–25 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Castiglione, G., Vaglica, R.: Recognizable Picture Languages and Polyominoes. In: Bozapalidis, S., Rahonis, G. (eds.) CAI 2007. LNCS, vol. 4728, pp. 160–171. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    De Carli, F., Frosini, A., Rinaldi, S., Vuillon, L.: On the Tiling System Recognizability of Various Classes of Convex Polyominoes. Ann. Comb. 13, 169–191 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Delest, M., Dubernard, J.P., Dutour, I.: Parallelogram Polyominoes and Corners. J. Symb. Comp. 20, 503–515 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Delest, M., Viennot, X.G.: Algebraic languages and polyominoes enumeration. Theoret. Comp. Sci. 34, 169–206 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Del Lungo, A., Duchi, E., Frosini, A., Rinaldi, S.: On the generation and enumeration of various classes of convex polyominoes. Electronic Journal of Combinatorics 11, 60 (2004)Google Scholar
  10. 10.
    Del Lungo, A., Mirolli, M., Pinzani, R., Rinaldi, S.: A Bijection for Directed-Convex Polyominoes. In: Proc. of DM-CCG 2001, Discrete Mathematics and Theoretical Computer Science AA, pp. 133–144 (2001)Google Scholar
  11. 11.
    Golomb, S.W.: Checker Boards and Polyominoes. The American Mathematical Monthly 61, 675–682 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuba, A., Balogh, E.: Reconstruction of convex 2D discrete sets in polynomial time. Theoret. Comp. Sci. 283(1), 223–242 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Massazza, P.: A CAT algorithm for sand piles. PU.M.A. 19(2-3), 147–158 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Massazza, P., Radicioni, R.: A CAT algorithm for the exhaustive generation of ice piles. RAIRO Theoretical Informatics and Applications 44, 525–543 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Massazza, P., Radicioni, R.: On the Exhaustive Generation of Symmetric Sand Piles. In Proc. of GASCom 2010, Montréal, September 2-4 (2010)Google Scholar
  16. 16.
    Ollinger, N.: Tiling the Plane with a Fixed Number of Polyominoes. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 638–647. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Pergola, E., Sulanke, R.A.: Schröder Triangles, Paths, and Parallelogram Polyominoes. Journal of Integer Sequences, 1 Article 98.1.7 (1998) Google Scholar
  18. 18.
    Polya, G.: On the number of certain lattice polygons. J. Comb. Theory 6, 102–105 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Roberto Mantaci
    • 1
  • Paolo Massazza
    • 2
  1. 1.LIAFA, CNRS UMR 7089Université Paris Diderot - Paris 7, Case 7014Paris Cedex 13France
  2. 2.Dipartimento di Informatica e ComunicazioneUniversità degli Studi dell’InsubriaVareseItaly

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