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

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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 
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     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},
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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

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