Geodesic
Signed Polyomino Tilings By $n$-in-Line Polyominoes and Gröbner Bases
Publications de l'Institut Mathématique, _N_S_99 (2016) no. 113, p. 31
Cet article a éte moissonné depuis la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Voir la notice de l'article

Conway and Lagarias observed that a triangular region $T(m)$ in a hexagonal lattice admits a \emph{signed tiling} by three-in-line polyominoes (tribones) if and only if $m\in\{9d-1,9d\}_{d\in\mathbb{N}}$. We apply the theory of Gröbner bases over integers to show that $T(m)$ admits a signed tiling by $n$-in-line polyominoes ($n$-bones) if and only if $ mı \{dn^2-1,dn^2\}_{dı\mathbb{N}}. $ Explicit description of the Gröbner basis allows us to calculate the `Gröbner discrete volume' of a lattice region by applying the division algorithm to its `Newton polynomial'. Among immediate consequences is a description of the \emph{tile homology group} for the $n$-in-line polyomino.
DOI : 10.2298/PIM1613031M
Classification : 52C20, 13P10
Keywords: signed polyomino tilings, Gröbner bases 
@article{10_2298_PIM1613031M,
     author = {Manuela Muzika Dizdarevi\'c and Marinko Timotijevi\'c and Rade  T.  \v{Z}ivaljevi\'c},
     title = {Signed {Polyomino} {Tilings} {By} $n${-in-Line} {Polyominoes} and {Gr\"obner} {Bases}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {31 },
     year = {2016},
     volume = {_N_S_99},
     number = {113},
     doi = {10.2298/PIM1613031M},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM1613031M/}
}
TY  - JOUR
AU  - Manuela Muzika Dizdarević
AU  - Marinko Timotijević
AU  - Rade  T.  Živaljević
TI  - Signed Polyomino Tilings By $n$-in-Line Polyominoes and Gröbner Bases
JO  - Publications de l'Institut Mathématique
PY  - 2016
SP  - 31 
VL  - _N_S_99
IS  - 113
UR  - http://geodesic.mathdoc.fr/articles/10.2298/PIM1613031M/
DO  - 10.2298/PIM1613031M
LA  - en
ID  - 10_2298_PIM1613031M
ER  - 
%0 Journal Article
%A Manuela Muzika Dizdarević
%A Marinko Timotijević
%A Rade  T.  Živaljević
%T Signed Polyomino Tilings By $n$-in-Line Polyominoes and Gröbner Bases
%J Publications de l'Institut Mathématique
%D 2016
%P 31 
%V _N_S_99
%N 113
%U http://geodesic.mathdoc.fr/articles/10.2298/PIM1613031M/
%R 10.2298/PIM1613031M
%G en
%F 10_2298_PIM1613031M
Manuela Muzika Dizdarević; Marinko Timotijević; Rade  T.  Živaljević. Signed Polyomino Tilings By $n$-in-Line Polyominoes and Gröbner Bases. Publications de l'Institut Mathématique, _N_S_99 (2016) no. 113, p. 31 . doi: 10.2298/PIM1613031M

Cité par Sources :