Ribbon tile invariants

Author:
Igor Pak

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

MSC (2000):
Primary 05E10, 52C20; Secondary 05B45, 20C30

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

Published electronically:
August 8, 2000

MathSciNet review:
1781275

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite set of tiles, and a set of regions tileable by . We introduce a *tile counting group* as a group of all linear relations for the number of times each tile can occur in a tiling of a region . 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:
https://doi.org/10.1090/S0002-9947-00-02666-0

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