Maximal degree subposets of \(\nu\)-Tamari lattices (Q6106300)

Summary: 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 abo ve \(\nu\). 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.