
The estimation of the number of $p2$tilings of a plane by a given area polyomino
A. V. Shutov^{a}, E. V. Kolomeykina^{bc} ^{a} Vladimir State University
^{b} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^{c} Bauman Moscow State Technical University
Abstract:
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
selfavoiding 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.
Keywords:
tilings, regular tilings, crystallographic groups, $p2$tilings, polyomino, selfavoiding walks.
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UDC:
514.174.5 Received: 12.06.2016 Accepted:13.09.2016
Citation:
A. V. Shutov, E. V. Kolomeykina, “The estimation of the number of $p2$tilings of a plane by a given area polyomino”, Chebyshevskii Sb., 17:3 (2016), 204–214
Citation in format AMSBIB
\Bibitem{ShuKol16}
\by A.~V.~Shutov, E.~V.~Kolomeykina
\paper The estimation of the number of $p2$tilings of a plane by a given area polyomino
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 3
\pages 204214
\mathnet{http://mi.mathnet.ru/cheb509}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3684271}
\elib{https://elibrary.ru/item.asp?id=27452094}
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