Maximal degree subposets of $\nu$-Tamari lattices

TL;DR

This paper investigates special subposets of the $ u$-Tamari lattice characterized by maximal in-degree and out-degree, revealing their structure and isomorphisms with other Dyck path posets, thus advancing understanding of $ u$-Tamari lattice properties.

Contribution

It introduces and characterizes two new subposets of the $ u$-Tamari lattice based on degree conditions and establishes their isomorphisms with related Dyck path posets.

Findings

Maximal in-degree/out-degree relate to staircase shape paths.

Maximal out-degree poset is isomorphic to a $(m-1)$-Dyck path lattice.

Maximal in-degree poset is isomorphic to a greedy-ordered $(m-1)$-Dyck path lattice.

Abstract

In this paper, we study two different subposets of the $\nu$-Tamari lattice: one in which all elements have maximal in-degree and one in which all elements have maximal out-degree. The maximal in-degree and maximal out-degree of a $\nu$-Dyck path turns out to be the size of the maximal staircase shape path that fits weakly above $\nu$. For $m$-Dyck paths of height $n$, we further show that the maximal out-degree poset is poset isomorphic to the $\nu$-Tamari lattice of $(m-1)$-Dyck paths of height $n$, and the maximal in-degree poset is poset isomorphic to the $(m-1)$-Dyck paths of height $n$ together with a greedy order. We show these two isomorphisms and give some properties on $\nu$-Tamari lattices along the way.