Rhombic tilings of polygons and classes of reduced words in Coxeter groups (Q1352860)

This paper presents and explores bijections between equivalence classes of reduced words, rhombic tilings of (possibly non-convex) polygons and lattice paths. In particular, Theorem 2.2 establishes, for every permutation \(\sigma\) of \(n\) elements, a bijection between the set of rhombic tilings of a certain \((2n)\)-gon \(X(\sigma)\) and the equivalence classes of reduced decompositions of \(\sigma\) modulo commuting non-adjacent transpositions. This extends a known correspondence in the special case when \(\sigma =w_0\) is the order-reversing permutation---the only case when \(X(\sigma)\) is convex. A further correspondence between rhombic tilings of certain centrally-symmetric 8-gons (of side lengths \(a,b,1,1\)) and lattice paths provides the tools for the proof of a formula for the number of such tilings, conjectured by Kuperberg and Propp, together with a \(q\)-analogue. Finally, extensions for signed permutations of type \(B_n\) and \(D_n\) are presented.