Maximal Degree Subposets of v-Tamari Lattices

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