Signed polyomino tilings by n-in-line polyominoes and Groebner bases
Combinatorics
2014-09-10 v1
Abstract
Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits signed tiling by three-in-line polyominoes (tribones) if and only if m=9d-1 or m=9d for some integer d. We apply the theory of Groebner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m=dn^2-1 or m=dn^2 for some integer d. Explicit description of the Groebner basis allows us to calculate the "Groebner discrete volume" of a lattice region by applying the division algorithm to its `Newton polynomial'. Among immediate consequences is a description of the tile homology group of the -in-line polyomino.
Keywords
Cite
@article{arxiv.1409.2745,
title = {Signed polyomino tilings by n-in-line polyominoes and Groebner bases},
author = {Manuela Muzika Dizdarević and Marinko Timotijević and Rade T. Živaljević},
journal= {arXiv preprint arXiv:1409.2745},
year = {2014}
}