The steep-bounce zeta map in parabolic Cataland

Wenjie Fang, Cesar Ceballos, Henri Mühle

Research output: Contribution to conferenceAbstractResearchpeer-review

Abstract

As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these general-izations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to ν-Tamari lattices for bounce paths ν. We then introduce a new combinatorialobject called “left-aligned colored tree”, and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the Steep-Bounce Conjecture using a generalization of the famous zeta map in q,t-Catalan combinatorics.
Original languageEnglish
Publication statusAccepted/In press - 2019

Fingerprint

Bounce
Bijective
Dyck Paths
Combinatorics
Isomorphic
Path

Keywords

  • parabolic Tamari lattice
  • ν-Tamari lattice
  • bijection
  • left-aligned colorable tree
  • zeta map

Cite this

Fang, W., Ceballos, C., & Mühle, H. (Accepted/In press). The steep-bounce zeta map in parabolic Cataland.

The steep-bounce zeta map in parabolic Cataland. / Fang, Wenjie; Ceballos, Cesar; Mühle, Henri.

2019.

Research output: Contribution to conferenceAbstractResearchpeer-review

Fang, W, Ceballos, C & Mühle, H 2019, 'The steep-bounce zeta map in parabolic Cataland'.
Fang W, Ceballos C, Mühle H. The steep-bounce zeta map in parabolic Cataland. 2019.
Fang, Wenjie ; Ceballos, Cesar ; Mühle, Henri. / The steep-bounce zeta map in parabolic Cataland.
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AB - As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these general-izations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to ν-Tamari lattices for bounce paths ν. We then introduce a new combinatorialobject called “left-aligned colored tree”, and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the Steep-Bounce Conjecture using a generalization of the famous zeta map in q,t-Catalan combinatorics.

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