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Smooth partitions and Chebyshev polynomials

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 ∈ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Smooth partitions and Chebyshev polynomials

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References (3)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdp071
Publisher site
See Article on Publisher Site

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.

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Dec 1, 2009

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