## 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 ∈ B_{s} implies that i + 1 ∈ B_{s-1} ∪ B_{s} ∪ B_{s+1} for all i ∈ [n - 1] (B_{0} = 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.

Original language | English |
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Pages (from-to) | 961-970 |

Number of pages | 10 |

Journal | Bulletin of the London Mathematical Society |

Volume | 41 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2009 |

## ASJC Scopus subject areas

- Mathematics (all)