Maximal degree subposets of ν-Tamari lattices

In this paper, we study two different subposets of the v-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 v-Dyck path turns out to be the size of the maximal staircase shape path that fits weakly above v. For m-Dyck paths of height n, we further show that the maximal out-degree poset is poset isomorphic to the v-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 v-Tamari lattices along the way.