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

Wenjie Fang

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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).
Original languageEnglish
Pages (from-to)1-30
Number of pages30
JournalAdvances in Applied Mathematics
Volume95
DOIs
Publication statusPublished - Apr 2018

Fingerprint

Planar Maps
Nonseparable
Duality
Interval
Bijection
Involution
Duality Maps
Bijective
Isomorphic
Closed
Class
Object

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.

In: Advances in Applied Mathematics, Vol. 95, 04.2018, p. 1-30.

Research output: Contribution to journalArticleResearchpeer-review

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