Séminaire Lotharingien de Combinatoire, 82B.36 (2019), 12 pp.
Cesar Ceballos, Wenjie Fang, and Henri Mühle
The Steep-Bounce Zeta Map in Parabolic Cataland
As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations 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 combinatorial object called "left-aligned colorable 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.
Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.
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