Smooth partitions and Chebyshev polynomials

Abstract

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 ∈ Bs implies that i + 1 ∈ Bs−1 ∪ Bs ∪ Bs+1 for all i ∈ [n − 1] (B0 = Bk + 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 2n − 1, we prove that the counting sequence for smooth partitions of [n] is Sloane's A005773.