## Abstract

How many ways can a rectangle be partitioned into smaller ones? We study two variants of this problem: when the partitions are constrained to lie on n given points (no two of which are corectilinear), and when there are no such constraints and all we require is that the number of (non-intersecting) segments is n. In the first case, when the order (permutation) of the points conforms with a certain property, the number of partitions is the (n + 1)st Baxter number, B(n + 1); the number of permutations conforming with the property is the (n - 1)st Schröder number; and the number of guillotine partitions is the nth Schröder number. In the second case, it is known that the number of partitions and the number of guillotine partitions correspond to the Baxter and Schröder numbers, respectively. Our contribution is a bijection between permutations and partitions. Our results provide interesting and new geometric interpretations to both Baxter and Schröder numbers and suggest insights regarding the intricacies of the interrelations.

Original language | English |
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Pages | 729-738 |

Number of pages | 10 |

State | Published - 2004 |

Externally published | Yes |

Event | Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms - New Orleans, LA., United States Duration: 11 Jan 2004 → 13 Jan 2004 |

### Conference

Conference | Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | New Orleans, LA. |

Period | 11/01/04 → 13/01/04 |

## Keywords

- Baxter permutations
- Guillotine partitions
- Quasimonotone permutations
- Rectangular partitions
- Schröder numbers

## ASJC Scopus subject areas

- Software
- General Mathematics