Signed tilings by ribbon L n-ominoes, n even, via Groebner bases

TL;DR

This paper characterizes when rectangles can be signed tiled by specific ribbon L-shaped n-ominoes using Groebner bases, revealing new algebraic conditions for tiling possibilities.

Contribution

It introduces algebraic methods with Groebner bases to determine signed tiling conditions for rectangles by ribbon L-ominoes, extending previous combinatorial results.

Findings

Signed tilings require sides to be even with divisibility conditions

Explicit Groebner bases are constructed for tiling ideals

Results extend previous tiling theorems beyond coloring invariants

Abstract

Let $\mathcal{T}_n$ be the set of ribbon $L$-shaped $n$-ominoes for some $n\ge 4$ even, and let $\mathcal{T}_n^+$ be $\mathcal{T}_n$ with an extra $2\times 2$ square. We investigate signed tilings of rectangles by $\mathcal{T}_n$ and $\mathcal{T}_n^+$. We show that a rectangle has a signed tiling by $\mathcal{T}_n$ if and only if both sides of the rectangle are even and one of them is divisible by $n$, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ We also show that a rectangle has a signed tiling by $\mathcal{T}_n^+, n\ge 6$ even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ Our proofs are based on the exhibition of explicit Gr\"obner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in \emph{ V.~Nitica, Every tiling of the first quadrant by ribbon $L$ $n$-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, {\em 5}, (2015) 11--25,} cannot be obtained from coloring invariants.