Ribbon tile invariants

Pak, Igor

doi:10.1090/S0002-9947-00-02666-0

Ribbon tile invariants
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by Igor Pak

Trans. Amer. Math. Soc. 352 (2000), 5525-5561

DOI: https://doi.org/10.1090/S0002-9947-00-02666-0

Published electronically: August 8, 2000

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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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