Ribbon tile invariants from the signed area
type: Item
instance of: scholarly article
title: Ribbon tile invariants from the signed area
zbMATH Open document ID: 1001.05040
author: Igor Pak
author: Cristopher Moore
published in: Journal of Combinatorial Theory. Series A
publication date: 2002-12-11T00:00:00Z
review text: Ribbon tiles are polyominoes consisting of \(n\) squares of the square grid laid out along a path which, at each step, moves north or east. A ribbon tile invariant is a translation-invariant mapping \(\varphi\) on the family of ribbon tiles such that, if \(\Gamma\) is a simply-connected polyomino and \(\nu\) a tiling of \(\Gamma\) by ribbon tiles \(\tau\), then \(\sum_{\tau\in \nu} \varphi(\tau)\) depends only on \(\Gamma\) but not on \(\nu\). The authors establish the existence of a large class of ribbon tile invariants. In particular, they prove a conjecture about a full basis of ribbon tile invariants, made in \textit{I. Pak} [Trans. Am. Math. Soc. 352, 5525-5561 (2000; Zbl 0963.05030)]. Their results are applied to determine the structure of the tile counting group for ribbon tiles introduced in Pak's paper mentioned above.
reviewed by: Egon Schulte
Mathematics Subject Classification ID: 05B50
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 52C20
zbMATH DE Number: 1760663
zbMATH Keywords: polyomino tilings
zbMATH Keywords: tile invariants
zbMATH Keywords: height representation
MaRDI profile type: Publication
OpenAlex ID: W2016296196
cites work: Tiling with polyominoes and combinatorial group theory
cites work: Q4348479
cites work: Q4228435
cites work: A note on tiling with integer-sided rectangles
cites work: Q4231920
cites work: Hard tiling problems with simple tiles
cites work: On tilings by ribbon tetrominoes
cites work: Ribbon tile invariants
cites work: A Pedestrian Approach to a Method of Conway, or, A Tale of Two Cities
cites work: Exact sampling with coupled Markov chains and applications to statistical mechanics
cites work: Tiling groups: New applications in the triangular lattice
cites work: Conway's Tiling Groups
DOI: 10.1006/JCTA.2001.3208
description: scientific article; zbMATH DE number 1760663