Séminaire Lotharingien de Combinatoire, 82B.36 (2019), 12 pp.

Cesar Ceballos, Wenjie Fang, and Henri Mühle

The Steep-Bounce Zeta Map in Parabolic Cataland

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 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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