EMIS ELibM Electronic Journals Publications de l'Institut Mathématique, Nouvelle Série
Vol. 99(113), pp. 31–42 (2016)

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Manuela Muzika Dizdarevic, Marinko Timotijevic, Rade T. Zivaljevic

Faculty of Natural Sciences and Mathematics, Sarajevo, Bosnia and Herzegovina; Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Serbia; Mathematical Institute SASA, Belgrade, Serbia

Abstract: 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 $$ m\in \{dn^2-1,dn^2\}_{d\in\mathbb{N}}. $$ 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.

Keywords: signed polyomino tilings; Gröbner bases

Classification (MSC2000): 52C20; 13P10

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Electronic fulltext finalized on: 12 Apr 2016. This page was last modified: 20 Apr 2016.

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