Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups

AbstractIn the standard Coxeter presentation, the symmetric groupSnis generated by the adjacent transpositions (1, 2), (2, 3), …, (n−1, n). For any given permutation, we consider all minimal-length factorizations thereof as a product of the generators. Any two transpositions (i, i+1) and (j, j+1) commute if the numbersiandjare not consecutive; thus, in any factorization, their order can be switched to obtain another factorization of the same permutation. Extending this to an equivalence relation, we establish a bijection between the resulting equivalence classes and rhombic tilings of a certain 2n-gon determined by the permutation. We also study the graph structure induced on the set of tilings by the other Coxeter relations. For a special case, we use lattice-path diagrams to prove an enumerative conjecture by Kuperberg and Propp, as well as aq-analogue thereof. Finally, we give similar constructions for two other families of finite Coxeter groups, namely those of typesBandD

Finite Coxeter Groups and Generalized Elnitsky Tilings

In [5], Elnitsky constructed three elegant bijections between classes of reduced words for Type $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{D}$ families of Coxeter groups and certain tilings of polygons. This paper offers a particular generalization of this concept to all finite Coxeter Groups in terms of embeddings into the Symmetric Group. [5] Elnitsky, Serge. Rhombic tilings of polygons and classes of reduced words in Coxeter groups. PhD dissertation, University of Michigan, 1993

A formula for the number of tilings of an octagon by rhombi

We propose the first algebraic determinantal formula to enumerate tilings of a centro-symmetric octagon of any size by rhombi. This result uses the Gessel-Viennot technique and generalizes to any octagon a formula given by Elnitsky in a special case.Comment: New title. Minor improvements. To appear in Theoretical Computer Science, special issue on "Combinatorics of the Discrete Plane and Tilings

Enumeration of octagonal tilings

Random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms. Of particular interest is the tiling entropy density, which measures the relation of the number of distinct tilings to the number of constituent tiles. Tilings by squares and 45 degree rhombi receive special attention as presumably the simplest model that has not yet been solved exactly in the thermodynamic limit. However, an exact enumeration formula can be evaluated for tilings in finite regions with fixed boundaries. We implement this algorithm in an efficient manner, enabling the investigation of larger regions of parameter space than previously were possible. Our new results appear to yield monotone increasing and decreasing lower and upper bounds on the fixed boundary entropy density that converge toward S = 0.36021(3)