The estimating of the number of lattice tilings of a plane by a given area centrosymmetrical polyomino

A. V. Shutov ; E. V. Kolomeykina

Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 2, pp. 295-303

Résumé

We study a problem about the number of lattice plane tilings by the given area centrosymmetrical polyominoes. A polyomino is a connected plane geomatric figure formed by joiining a finite number of 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 plane by polyominoes. In particular, the tiling is a lattice tiling if each tile can be mapped to any other tile by a translation which maps the whole tiling to itself. Earlier we proved that, for the number $T(n)$ of a lattice plane tilings by polyominoes of an area $n$, holds the inequalities $2^{n-3}+2^{[\frac{n-3}{2}]}\leq T(n)\leq C(n+1)^3 (2,7)^{n+1}$. In the present work we prove a similar estimate for the number of lattice tilings with an additional central symmetry. Let $T_c(n)$ be a number of lattice plane tilings by a given area centrosymmetrical polyominoes such that its translation lattice is a sublattice of $\mathbb{Z}^2$. It is proved that $C_1(\sqrt 2)^n\leq T_c(n) \leq C_2n^2(\sqrt{2.68})^n$. In the proof of a lower bound we give an explicit construction of required lattice plane tilings. The proof of an upper bound is based on a criterion of the existence of lattice plane tiling by polyominoes, and on the theory of self-avoiding walks on a square lattice.

Détail

Keywords: tilings, polyomino.