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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ribbon tile invariants


Author: Igor Pak
Journal: Trans. Amer. Math. Soc. 352 (2000), 5525-5561
MSC (2000): Primary 05E10, 52C20; Secondary 05B45, 20C30
Published electronically: August 8, 2000
MathSciNet review: 1781275
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Abstract: Let $\mathbf{T}$ be a finite set of tiles, and $\mathcal{B}$ a set of regions $\Gamma $ tileable by $\mathbf{T}$. We introduce a tile counting group $\mathbb{G} (\mathbf{T}, \mathcal{B})$ as a group of all linear relations for the number of times each tile $\tau \in \mathbf{T}$ can occur in a tiling of a region $\Gamma \in \mathcal{B}$. We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group.

The tile counting group is presented by its set of generators, which consists of certain new tile invariants. In a special case these invariants generalize the Conway-Lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group.

The heart of the proof is the known bijection between rim hook tableaux and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.


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Additional Information

Igor Pak
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520-8283
Address at time of publication: Department of Mathematics, MIT, Cambridge, Massachusetts 02139
Email: paki@math.yale.edu, paki@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02666-0
PII: S 0002-9947(00)02666-0
Keywords: Polyomino tilings, tile invariants, Conway group, rim (ribbon) hooks, Young diagrams, Young tableaux, rim hook bijection, symmetric group
Received by editor(s): December 12, 1997
Published electronically: August 8, 2000
Article copyright: © Copyright 2000 American Mathematical Society