## Abstract

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]: Suppose that the subsets B_{i} are listed in increasing order of their minimal elements and π = π_{1}π_{2}···π_{n} denotes the canonical sequential form of a partition of [n] in which i ε B_{πi} for each i: In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of π between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of π of the same letter lying between two letters which are strictly larger.

Original language | English |
---|---|

Pages (from-to) | 284-308 |

Number of pages | 25 |

Journal | Applicable Analysis and Discrete Mathematics |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2010 |

## Keywords

- Partition statistics
- Set partition
- Stirling number

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics