Reduced words in affine Coxeter groups

Let 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 A,,, and he suggested looking at r(w) for the ffie group Aln. We prove that for any afline Coxeter group R, there is a finite number of types of elements in xX, such that to every element w can be associated (I) a type t, (2) an element u in the &rite group X,, and (3) an n-tuple (mt,mz,...,m,) of integers rn: 2 0. Then r(w) = $(ml, . . . , m,), and for every r: and for large enough mi, a homogeneous linear ~~rnen~o~ recurrence holds. For A”,, this takes a nice cornbi~to~~ form. We also discuss a canonical reduced word for w associated to its n-tuple.