  An element of a Coxeter group $W$ is fully commutative if any two of its
reduced decompositions are related by a series of transpositions of adjacent
commuting generators. In the present work, we focus on fully commutative
involutions, which are characterized in terms of Viennot's heaps. By encoding
the latter by Dyck-type lattice walks, we enumerate fully commutative
involutions according to their length, for all classical finite and affine
Coxeter groups. In the finite cases, we also find explicit expressions for
their generating functions with respect to the major index. Finally in affine
type $A$, we connect our results to Fan--Green's cell structure of the
corresponding Temperley--Lieb algebra.

Combinatorics