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Polyominoes Determined by Permutations: Enumeration via Bijections

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Abstract

A permutominide is a set of cells in the plane satisfying special connectivity constraints and uniquely defined by a pair of permutations. It naturally generalizes the concept of permutomino, recently investigated by several authors and from different points of view [1, 2, 4, 6, 7]. In this paper, using bijective methods, we determine the enumeration of various classes of convex permutominides, including, parallelogram, directed convex, convex, and row convex permutominides. As a corollary we have a bijective proof for the number of convex permutominoes, which was still an open problem.

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References

  1. Bernini, A., Disanto, F., Pinzani, R., Rinaldi, S.: Permutations defining convex permutominoes. J. Integer Seq. 10, Article 07.9.7 (2007)

  2. Boldi P., Lonati V., Radicioni R., Santini M.: The number of convex permutominoes. Inform. and Comput. 206(9-10), 1074–1083 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bousquet-Mèlou M.: A method for the enumeration of various classes of column convex polygons. Discrete Math. 154(1-3), 1–25 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Disanto F., Frosini A., Pinzani R., Rinaldi S.: A closed formula for the number of convex permutominoes. Electron. J. Combin. 14(1), #R57 (2007)

    MathSciNet  Google Scholar 

  5. Duchi, E., Poulalhon, D.: On square permutations. Discrete Math. Theor. Comput. Sci. Proc. 207–222 (2008)

  6. Incitti F.: Permutation diagrams, fixed points and Kazdhan-Lusztig R-polynomials. Ann. Combin. 10(3), 369–387 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kassel C., Lascoux A., Reutenauer C.: The singular locus of a Schubert variety. J. Algebra 269(1), 74–108 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche e Informatiche, Università di Siena, Pian dei Mantellini 44, 53100, Siena, Italy

    Filippo Disanto & Simone Rinaldi

  2. Université Paris 7, LIAFA, Denis Diderot, place Jussieu, F-75251, Paris Cedex 05, France

    Enrica Duchi

  3. Dipartimento di Sistemi e Informatica, Università degli Studi di Firenze, Viale Morgagni 65, 50134, Firenze, Italy

    Renzo Pinzani

Authors
  1. Filippo Disanto
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  2. Enrica Duchi
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  3. Renzo Pinzani
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  4. Simone Rinaldi
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Corresponding author

Correspondence to Filippo Disanto.

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Disanto, F., Duchi, E., Pinzani, R. et al. Polyominoes Determined by Permutations: Enumeration via Bijections. Ann. Comb. 16, 57–75 (2012). https://doi.org/10.1007/s00026-011-0121-6

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  • DOI: https://doi.org/10.1007/s00026-011-0121-6

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