# The Steep-Bounce Zeta Map in Parabolic Cataland

## TL;DR

This paper unifies various Tamari lattice generalizations through bijections, introduces new combinatorial objects, and proves the Steep-Bounce Conjecture by generalizing the zeta map in $q,t$-Catalan theory.

## Contribution

It establishes isomorphisms between parabolic and $
u$-Tamari lattices, introduces left-aligned colorable trees, and proves the Steep-Bounce Conjecture via a generalized zeta map.

## Key findings

- Parabolic Tamari lattices are isomorphic to $
u$-Tamari lattices.
- Introduces left-aligned colorable trees as a bijective tool.
- Proves the Steep-Bounce Conjecture using a generalized zeta map.

## Abstract

As a classical object, the Tamari lattice has many generalizations, including $\nu$-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 $\nu$-Tamari lattices for bounce paths $\nu$. 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. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection.