Combinatorial aspects of L-convex polyominoes

  • Authors:

  • G. Castiglione;A. Frosini;E. Munarini;A. Restivo;S. Rinaldi

  • Affiliations:

  • Universití di Palermo, Dipartimento di Matematica e Applicazioni, Via Archirafi, 34, 90123, Palermo, Italy;Universití di Siena, Dipartimento di Matematica, Pian dei Mantellini, 44, 53100, Siena, Italy;Politecnico di Milano, Dipartimento di Matematica, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy;Universití di Palermo, Dipartimento di Matematica e Applicazioni, Via Archirafi, 34, 90123, Palermo, Italy;Universití di Siena, Dipartimento di Matematica, Pian dei Mantellini, 44, 53100, Siena, Italy

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

We consider the class of L-convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an ''L'' shaped path in one of its four cyclic orientations. The paper proves bijectively that the number f"n of L-convex polyominoes with perimeter 2(n+2) satisfies the linear recurrence relation f"n"+"2=4f"n"+"1-2f"n, by first establishing a recurrence of the same form for the cardinality of the ''2-compositions'' of a natural number n, a simple generalization of the ordinary compositions of n. Then, such 2-compositions are studied and bijectively related to certain words of a regular language over four letters which is in turn bijectively related to L-convex polyominoes. In the last section we give a solution to the open problem of determining the generating function of the area of L-convex polyominoes.