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

TL;DR

This paper characterizes when rectangles can be signed tiled by ribbon L n-ominoes with odd n using Groebner bases, revealing conditions for signed and regular tilings and their obstructions.

Contribution

It introduces an explicit Groebner basis approach to determine signed tilings by ribbon L n-ominoes for rectangles and general regions, including obstructions.

Findings

A rectangle can be signed tiled by ribbon L n-ominoes if and only if a side is divisible by n.

Any k-inflated skewed L n-omino can be signed tiled by skewed L n-ominoes.

Obstructions to regular tilings exist beyond those detectable by signed tilings.

Abstract

We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Groebner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.