Tiling the Plane with Permutations

  • Alexandre Blondin Massé
  • Andrea Frosini
  • Simone Rinaldi
  • Laurent Vuillon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)


A permutomino is a polyomino uniquely determined by a pair of permutations. Recently permutominoes, and in particular convex permutominoes have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2,3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is called pseudo-square if its boundary word may be factorized as \(XY\overline{X} \,\overline{Y}\).

In this paper we consider the pseudo-square polyominoes which are also convex permutominoes. By using the Beauquier-Nivat characterization we provide some geometrical and combinatorial properties of such objects, and we show for any fixed X, each word Y such that \(XY\overline{X} \,\overline{Y}\) is pseudo-square is prefix of an infinite word Y  ∞  with period 4 |X| N |X| E .

Some conjectures obtained through exhaustive search are also presented and discussed in the final section.


Vertical Segment Tomographical Reconstruction Schubert Variety Boundary Word Leftmost Column 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alexandre Blondin Massé
    • 1
  • Andrea Frosini
    • 2
  • Simone Rinaldi
    • 3
  • Laurent Vuillon
    • 1
  1. 1.Laboratoire de MathématiquesUniversité de Savoie, CNRS UMR 5127Le Bourget du LacFrance
  2. 2.Dipartimento di Sistemi e InformaticaUniversità di FirenzeFirenzeItaly
  3. 3.Dipartimento di Matematica e InformaticaUniversità di SienaSienaItaly

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