The estimation of the number of $p2$-tilings of a plane by a given area polyomino

A. V. Shutov ; E. V. Kolomeykina

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 204-214

Résumé

We consider the problem about a number of $p2$-tilings of a plane by a given area polyominoes. A polyomino is a connected plane geometric figure formed by joining one or more unit squares edge to edge. At present, various combinatorial enumeration problems connected to the polyomino are actively studied. There are some interesting problems on enuneration of various classes of polyominoes and enumeration of tilings of finite regions or a whole plane by polyominoes. The tiling is called $p2$-tiling, if each tile can be mapped to any other tile by the translation or the central symmetry, and this transformation maps the whole tiling to itself. $p2$-tilings are special case of regular plane tilings. Let $t(n)$ be a number of $p2$-tilings of a plane by a $n$-area polyomino such that the lattices of periods of these tilings are sublattices of $\mathbb{Z}^2$. It is proved that following inequality is true: $ C_12^n \leq t(n)\leq C_2n^4(2.68)^n$. To prove the lower bound we use the exact construction of required tilings. The proof of the upper bound is based on the Conway criterion of the existence of $p2$-tilings of a plane. Also, the upper bound depends on the theory of self-avoiding walks on the square lattice. Earlier similar results were obtained by authors for the number of lattice tilings of a plane by a given area polyomino (it's more simple type of a plane tilings by polyomino), and for the number of lattice tilings of the plane by centrosimmetrical polyomino. Bibliography: 28 titles.

Détail

Keywords: tilings, regular tilings, crystallographic groups, $p2$-tilings, polyomino, self-avoiding walks.