From generalized Tamari intervals to non-separable planar maps

From generalized Tamari intervals to non-separable planar maps

Let v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM...

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Bibliographic Details
Main Authors: Wenjie Fang, Louis-François Préville-Ratelle
Format: Article
Language:English
Published: Discrete Mathematics & Theoretical Computer Science 2020-04-01
Series:Discrete Mathematics & Theoretical Computer Science
Subjects:
[math.math-co]mathematics [math]/combinatorics [math.co]
Online Access:https://dmtcs.episciences.org/6421/pdf
Description
Summary:Let v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM(v) is isomorphic to the dual of TAM(←−v ), where ←−v is the reverse of v with N and E exchanged. Our main contribution is a bijection from intervals in TAM(v) to non-separable planar maps. It follows that the number of intervals in TAM(v) over all v of length n is 2(3n+3)! (n+2)!(2n+3)! . This formula was first obtained by Tutte(1963) for non-separable planar maps.
ISSN:1365-8050

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