Ribbon tile invariants

Let T be a finite set of tiles, and B a set of regions F tileable by T. We introduce a tile counting group G(T, B) as a group of all linear relations for the number of times each tile T C T can occur in a tiling of a region r C 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.

[1]  Dennis E. White,et al.  A Schensted Algorithm for Rim Hook Tableaux , 1985, J. Comb. Theory, Ser. A.

[2]  P. Steerenberg,et al.  Targeting pathophysiological rhythms: prednisone chronotherapy shows sustained efficacy in rheumatoid arthritis. , 2010, Annals of the rheumatic diseases.

[3]  G. B. Robinson,et al.  Representation theory of the symmetric group , 1961 .

[4]  J. Propp,et al.  Local statistics for random domino tilings of the Aztec diamond , 1996, math/0008243.

[5]  Richard W. Kenyon.  A Note on Tiling with Integer-Sided Rectangles , 1996, J. Comb. Theory, Ser. A.

[6]  P. W. Kasteleyn.  The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice , 1961 .

[7]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[8]  Igor Pak,et al.  On Tilings by Ribbon Tetrominoes , 1999, J. Comb. Theory, Ser. A.

[9]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[10]  Greg Kuperberg,et al.  Alternating-Sign Matrices and Domino Tilings (Part II) , 1992 .

[11]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[12]  J. Propp,et al.  Alternating sign matrices and domino tilings , 1991, math/9201305.

[13]  Jeffrey C. Lagarias,et al.  Tiling with polyominoes and combinatorial group theory , 1990, J. Comb. Theory, Ser. A.

[14]  M. Fisher,et al.  Dimer problem in statistical mechanics-an exact result , 1961 .

[15]  Michelle L. Wachs,et al.  Generalized quotients in Coxeter groups , 1988 .

[16]  James Propp.  A pedestrian approach to a method of Conway, or, a tale of two cities , 1997 .

[17]  W. Thurston.  Conway's tiling groups , 1990 .