A bijection for the total area of parallelogram polyominoes

  • Authors:

  • Alberto Del Lungo;Maurice Nivat;Renzo Pinzani;Simone Rinaldi

  • Affiliations:

  • Dipartimento di Matematica, Università di Siena, Pian dei Mantellini 44, 53100, Siena, Italy;LIAFA, Université Denis Diderot 2, place Jussieu 75251 Paris Cedex 05, France;Dipartimento di Sistemi e Informatica, Via Lombroso 6/17, 50134 Firenze, Italy;Dipartimento di Matematica, Università di Siena, Pian dei Mantellini 44, 53100, Siena, Italy

  • Venue:
  • Discrete Applied Mathematics - Fun with algorithms 2 (FUN 2001)
  • Year:
  • 2004

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Abstract

The sum of the areas of the parallelogram polyominoes having semi-perimeter n + 2 is equal to 4n. In this paper we give a simple proof of this property by means of a mapping from the cells of parallelogram polyominoes having semiperimeter n + 2 to the 4n words of length n of the free monoid {a, b, c, d}*. This mapping works in linear time. Then, we introduce a tiling game arising from this enumerative property.