Smooth partitions and Chebyshev polynomials

A partition of the set [ n ]={1, 2, …, n } is a collection { B 1 , …, B k } 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] ( B 0 = B k + 1 = ∅). 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 2 n − 1, we prove that the counting sequence for smooth partitions of [ n ] is Sloane's A005773.