Ribbon tile invariants
Author:
Igor Pak
Journal:
Trans. Amer. Math. Soc. 352 (2000), 55255561
MSC (2000):
Primary 05E10, 52C20; Secondary 05B45, 20C30
Published electronically:
August 8, 2000
MathSciNet review:
1781275
Fulltext PDF Free Access
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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 ConwayLagarias 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 065208283
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/S0002994700026660
PII:
S 00029947(00)026660
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
