Length enumeration of fully commutative elements in finite and affine Coxeter groups

An element w of a Coxeter group W is said to be fully commutative if any reduced expression of w can be obtained from any other by a sequence of transpositions of adjacent commuting generators. These elements were described in 1996 by Stembridge in the case of finite irreducible groups, and more recently by Biagioli, Jouhet and Nadeau (BJN) in the affine cases. We focus here on the length enumeration of these elements. Using a recursive description, BJN established systems of non-linear q-equations for the associated generating functions. Here, we show that an alternative recursive description leads to explicit expressions for these generating functions.