Smooth partitions and Chebyshev polynomials

A partition of the set [n]=1, 2, ..., n is a collection B1, ..., Bk of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. A partition of [n] is said to be smooth if i ∈ B_s implies that i + 1 ∈ B_s-1 ∪ B_s ∪ B_s+1 for all i ∈ [n - 1] (B₀ = B_k+1 = φ). This paper presents the generating function for the number of k-block, smooth partitions of [n], written in terms of Chebyshev polynomials of the second kind. There follows a formula for the number of k-block, smooth partitions of [n] written in terms of trigonometric sums. Also, by first establishing a bijection between the set of smooth partitions of [n] and a class of symmetric Dyck paths of semilength 2n - 1, we prove that the counting sequence for smooth partitions of [n] is Sloane's A005773.