Restricted involutions and Motzkin paths

Given a permutation σ ∈ Sn, one can partition the set {1, 2, . . . , n} into intervals A1, . . . , At such that σ(Aj) = Aj for every j. The restrictions of σ to the intervals in the finest of these decompositions are called connected components of σ. A permutation σ with a single connected component is called connected. Given a permutation σ ∈ Sn, we define the reverse of σ to be the permutation σr = σψ, where ψ ∈ Sn is defined by ψ(i) = n + 1 − i. Similarly, the complement of σ is the permutation σc = ψσ and the reverse-complement of σ is the permutation σrc = ψσψ.