### Abstract

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

Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Advances in Applied Mathematics |

Volume | 95 |

DOIs | |

Publication status | Published - Apr 2018 |

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### Keywords

- non-separable planar maps
- β-(1,0) trees
- synchronized intervals
- bijection
- map duality
- recursive decomposition

### Cite this

**A trinity of duality: Non-separable planar maps, β(1,0)-trees and synchronized intervals.** / Fang, Wenjie.

Research output: Contribution to journal › Article › Research › peer-review

}

TY - JOUR

T1 - A trinity of duality: Non-separable planar maps, β(1,0)-trees and synchronized intervals

AU - Fang, Wenjie

PY - 2018/4

Y1 - 2018/4

N2 - The dual of a map is a fundamental construction on combinatorial maps, but many other combinatorial objects also possess their notion of duality. For instance, the Tamari lattice is isomorphic to its order dual, which induces an involution on the set of so-called “synchronized intervals” introduced by Préville-Ratelle and the present author. Another example is the class of β(1,0)-trees, which has a mysterious involution h proposed by Claesson, Kitaev and Steingrímsson (2009). These two classes of combinatorial objects are all in bijection with the class of non-separable planar maps, which is closed under map duality. In this article, we show that we can identify the notions of duality in these three classes using previously known natural bijections, which leads to a bijective proof of a result from Kitaev and de Mier (2013).

AB - The dual of a map is a fundamental construction on combinatorial maps, but many other combinatorial objects also possess their notion of duality. For instance, the Tamari lattice is isomorphic to its order dual, which induces an involution on the set of so-called “synchronized intervals” introduced by Préville-Ratelle and the present author. Another example is the class of β(1,0)-trees, which has a mysterious involution h proposed by Claesson, Kitaev and Steingrímsson (2009). These two classes of combinatorial objects are all in bijection with the class of non-separable planar maps, which is closed under map duality. In this article, we show that we can identify the notions of duality in these three classes using previously known natural bijections, which leads to a bijective proof of a result from Kitaev and de Mier (2013).

KW - non-separable planar maps

KW - β-(1,0) trees

KW - synchronized intervals

KW - bijection

KW - map duality

KW - recursive decomposition

U2 - 10.1016/j.aam.2017.10.001

DO - 10.1016/j.aam.2017.10.001

M3 - Article

VL - 95

SP - 1

EP - 30

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

ER -