2,107 research outputs found
Tilings of an Isosceles Triangle
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union
of N trianglescongruent to T, overlapping only at their boundaries. The
triangle T is the "tile". The tile may or may not be similar to ABC. In this
paper we study the case of isosceles (but not equilateral) ABC. We study three
possible forms of the tile: right-angled, or with one angle double another, or
with a 120 degree angle. In the case of a right-angled tile, we give a complete
characterization of the tilings, for N even, but leave open whether N can be
odd. In the latter two cases we prove the ratios of the sides of the tile are
rational, and give a necessary condition for the existence of an N-tiling. For
the case when the tile has one angle double another, we prove N cannot be prime
or twice a prime.Comment: 34 pages, 18 figures. This version supplies corrections and
simplification
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
Proof of two conjectures of Zuber on fully packed loop configurations
Two conjectures of Zuber [``On the counting of fully packed loops
configurations. Some new conjectures,'' preprint] on the enumeration of
configurations in the fully packed loop model on the square grid with periodic
boundary conditions, which have a prescribed linkage pattern, are proved.
Following an idea of de Gier [``Loops, matchings and alternating-sign
matrices,'' Discrete Math., to appear], the proofs are based on bijections
between such fully packed loop configurations and rhombus tilings, and the
hook-content formula for semistandard tableaux.Comment: 20 pages; AmS-LaTe
Fast domino tileability
Domino tileability is a classical problem in Discrete Geometry, famously
solved by Thurston for simply connected regions in nearly linear time in the
area. In this paper, we improve upon Thurston's height function approach to a
nearly linear time in the perimeter.Comment: Appeared in Discrete Comput. Geom. 56 (2016), 377-39
Tiles and colors
Tiling models are classical statistical models in which different geometric
shapes, the tiles, are packed together such that they cover space completely.
In this paper we discuss a class of two-dimensional tiling models in which the
tiles are rectangles and isosceles triangles. Some of these models have been
solved recently by means of Bethe Ansatz. We discuss the question why only
these models in a larger family are solvable, and we search for the Yang-Baxter
structure behind their integrablity. In this quest we find the Bethe Ansatz
solution of the problem of coloring the edges of the square lattice in four
colors, such that edges of the same color never meet in the same vertex.Comment: 18 pages, 3 figures (in 5 eps files
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