A polyominoes-permutations injection and tree-like convex polyominoes

  • Authors:

  • Gadi Aleksandrowicz;Andrei Asinowski;Gill Barequet

  • Affiliations:

  • Center for Graphics and Geometric Computing, Dept. of Computer Science, The Technion-Israel Inst. of Technology, Haifa 32000, Israel;Dept. of Mathematics, The Technion-Israel Inst. of Technology, Haifa 32000, Israel;Center for Graphics and Geometric Computing, Dept. of Computer Science, The Technion-Israel Inst. of Technology, Haifa 32000, Israel

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2012

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Abstract

Plane polyominoes are edge-connected sets of cells on the orthogonal lattice Z^2, considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [n], and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence a"n=6a"n"-"1-14a"n"-"2+16a"n"-"3-9a"n"-"4+2a"n"-"5, and compute the closed-form formula a"n=2^n^+^2-(n^3-n^2+10n+4)/2.