## Abstract

Suppose that the elements within each block of a partition π of [n]={1,2,.,n} are written in ascending order. By a parity succession, we will mean a pair of adjacent elements x and y within some block of π such that x≡y(mod2). Here, we consider the problem of counting the partitions of [n] according to the number of successions, extending recent results concerning successions on subsets and permutations. Using linear algebra, we determine a formula for the generating function which counts partitions having a fixed number of blocks according to size and number of successions. Furthermore, a special case of our formula yields an explicit recurrence for the generating function which counts the parity-alternating partitions of [n], i.e., those that contain no successions.

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

Pages (from-to) | 2642-2650 |

Number of pages | 9 |

Journal | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 9 |

DOIs | |

State | Published - 1 Nov 2013 |

## Keywords

- Generating function
- Parity succession
- Set partition
- Tridiagonal matrix

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics