Maximal degree subposets of $\nu$-Tamari lattices

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.