Geodesic
Gröbner Lattice-Point Enumerators and Signed Tiling by $k$-in-line Polyominoes
Kragujevac Journal of Mathematics, Tome 49 (2025) no. 3, p. 443

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Conway and Lagarias observed that a triangular region $T_2(n)$ in a hexagonal lattice admits a {\em signed tiling} by 3-in-line polyominoes (tribones) if and only if $n\in \{3^2d-1, 3^2d\}_{d\in \mathbb{N}}$. We apply the theory of Gröbner bases over integers to show that $T_3(n)$, a three dimensional lattice tetrahedron of edge-length $n$, admits a signed tiling by tribones if and only if $ n \in \{3^3d-2, 3^3d-1, 3^3d\}_{d\in \mathbb{N}}$. More generally we study \emph{Gröbner lattice-point enumerators} of lattice polytopes and show that they are (modular) quasipolynomials in the case of $k$-in-line polyominoes. As an example of the ``unusual cancelation phenomenon'', arising only in signed tilings, we exhibit a configuration of 15 tribones in the $3$-space such that exactly one lattice point is covered by an odd number of tiles.
Classification : 52C22, 05B45, 11P21, 13P10
Keywords: Signed polyomino tiling, Gröbner bases, Lattice-point enumerators 
Manuela Muzika Dizdarević; Marinko Timotijević; Rade T. Živaljević. Gröbner Lattice-Point Enumerators and Signed Tiling by $k$-in-line Polyominoes. Kragujevac Journal of Mathematics, Tome 49 (2025) no. 3, p. 443 . http://geodesic.mathdoc.fr/item/KJM_2025_49_3_a8/
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     title = {Gr\"obner {Lattice-Point} {Enumerators} and {Signed} {Tiling} by $k$-in-line {Polyominoes}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {443 },
     year = {2025},
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     url = {http://geodesic.mathdoc.fr/item/KJM_2025_49_3_a8/}
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