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. 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.

作为经典对象，Tamari格具有许多概括，包括ν -Tamari格和抛物线Tamari格。在本文中，我们以双射的方式统一这些概括。我们首先证明抛物线Tamari晶格与v -Tamari晶格对于反弹路径ν是同构的。然后，我们引入了一个称为“左对齐的可着色树”的新组合对象，并表明它在各种抛物线加泰罗尼亚对象和某些Dyck路径嵌套对之间提供了双射的桥梁。结果，我们使用了著名的zeta映射的推广来证明“陡弹跳猜想”$q，Ť$-加泰罗尼亚组合。停车函数的zeta映射的一般化也出现在对角谐波理论中，作为我们双射的标记形式。