Signed polyomino tilings by n-in-line polyominoes and Groebner bases

TL;DR

This paper extends the characterization of signed tilings of triangular regions in hexagonal lattices by n-in-line polyominoes using Groebner bases, providing explicit formulas and algebraic tools for analysis.

Contribution

It introduces a Groebner basis approach to determine when regions admit signed tilings by n-in-line polyominoes, generalizing previous results for tribones.

Findings

Characterization of signed tilings for n-in-line polyominoes

Explicit Groebner basis for tiling conditions

Calculation method for the Groebner discrete volume

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 $n$-in-line polyomino.