Reduced words in affine Coxeter groups

AbstractLet r(w) denote the number of reduced words for an element w in a Coxeter group W. Stanley proved a formula for r(w) when W is the symmetric group An, and he suggested looking at r(w) for the affine group An. We prove that for any affine Coxeter group Xn there is a finite number of types of elements in Xn, such that to every element w can be associated (1) a type t, (2) an element v in the finite group Xn, and (3) an n-tuple (m1,m2,…,mn) of integers mi ⩾ 0. Then r(w) = rvt(m1,…,mn), and for every rvt and for large enough mi, a homogeneous linear n-dimensional recurrence holds. For An, this takes a nice combinatorial form. We also discuss a canonical reduced word for w associated to its n-tuple