RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Model. Anal. Inform. Sist.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Model. Anal. Inform. Sist., 2013, Volume 20, Number 5, Pages 148–157 (Mi mais337)  

This article is cited in 3 scientific papers (total in 3 papers)

The Estimation of the Number of Lattice Tilings of a Plane by a Given Area Polyomino

A. V. Shutova, E. V. Kolomeykinab

a Vladimir State University, , Stroitelei str., 11, Vladimir, 600024, Russia
b Moscow State Technical University, 2-nd Bauman str., 5, Moscow, 105005, Russia

Abstract: We study a problem of a number of lattice plane tilings by given area polyominoes. A polyomino is a connected plane geometric figure formed by joining edge to edge a finite number of unit squares. A tiling is a lattice tiling if each tile can be mapped to any other tile by translation which maps the whole tiling to itself. Let $T(n)$ be a number of lattice plane tilings by given area polyominoes such that its translation lattice is a sublattice of $\mathbb{Z}^2$. It is proved that $2^{n-3}+2^{[\frac{n-3}{2}]}\leq T(n)\leq C(n+1)^3(2.7)^{n+1}$. 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 polyomino and on the theory of self-avoiding walk. Also, it is proved that almost all polyominoes that give lattice plane tilings have sufficiently large perimeters.

Keywords: tilings, polyomino.

Full text: PDF file (169 kB)
References: PDF file   HTML file
UDC: 514.174.5
Received: 21.10.2013

Citation: A. V. Shutov, E. V. Kolomeykina, “The Estimation of the Number of Lattice Tilings of a Plane by a Given Area Polyomino”, Model. Anal. Inform. Sist., 20:5 (2013), 148–157

Citation in format AMSBIB
\Bibitem{ShuKol13}
\by A.~V.~Shutov, E.~V.~Kolomeykina
\paper The Estimation of the Number of Lattice Tilings of a Plane by a Given Area Polyomino
\jour Model. Anal. Inform. Sist.
\yr 2013
\vol 20
\issue 5
\pages 148--157
\mathnet{http://mi.mathnet.ru/mais337}


Linking options:
  • http://mi.mathnet.ru/eng/mais337
  • http://mi.mathnet.ru/eng/mais/v20/i5/p148

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Shutov, E. V. Kolomeikina, “Otsenka chisla reshetchatykh razbienii ploskosti na tsentralno-simmetrichnye polimino zadannoi ploschadi”, Model. i analiz inform. sistem, 22:2 (2015), 295–303  mathnet  mathscinet  elib
    2. A. V. Shutov, E. V. Kolomeikina, “Otsenka chisla $p2$–razbienii ploskosti na polimino zadannoi ploschadi”, Chebyshevskii sb., 17:3 (2016), 204–214  mathnet  mathscinet  elib
    3. A. V. Shutov, E. V. Kolomeikina, “O chisle razbienii ploskosti na poligeksy”, Dalnevost. matem. zhurn., 17:2 (2017), 257–264  mathnet  elib
  • Моделирование и анализ информационных систем
    Number of views:
    This page:368
    Full text:47
    References:32

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020