On some posets and lattices with the same height

Abstract

For a finite poset $\mathcal{P}$, its height $h(\mathcal{P})$ is the number of cover relations in its longest chain. When $\mathcal{P}$ is a lattice $\mathcal{L}$, we label its elements $x$ with $h(x_{\downarrow })=h([\hat{0},x])$ and its cover relations $x⋖y$ with $h(y_{\downarrow })-h(x_{\downarrow })$. When a lattice ${\mathcal{L}}^{\prime }$ extends $\mathcal{L}$, $h(x_{\downarrow }{)}_{\mathcal{L}}\le h(x_{\downarrow }{)}_{{\mathcal{L}}^{\prime }}$. We study lattices $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ such that $h(x_{\downarrow }{)}_{\mathcal{L}}=h(x_{\downarrow }{)}_{{\mathcal{L}}^{\prime }}$. Cover relations labeled $1$ in $\mathcal{L}$ induce a poset that we call the (long) skeletal poset $\mathrm{SK}(\mathcal{L})$. Its Hasse diagram is the largest spanning subgraph that the Hasse diagrams of $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ have in common. An example of lattices $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ is the alt-Tamari lattices introduced by Chenevière, where every alt-Tamari lattice $\mathrm{alt}\text{-}{\mathrm{Tam}}_n$ extends the Tamari lattice ${\mathrm{Tam}}_n$/refines the Dyck lattice ${\mathrm{Dyck}}_n$ such that $h(x_{\downarrow }{)}_{{\mathrm{Tam}}_n}=h(x_{\downarrow }{)}_{\mathrm{alt}\text{-}{\mathrm{Tam}}_n}$. We study $\mathrm{SK}({\mathrm{Tam}}_n)$ with another poset we introduce. We enumerate intervals in these posets. For a well-chosen distributive lattice, we introduce its altitude lattices, which generalize the alt-Tamari lattices $\mathrm{alt}\text{-}{\mathrm{Tam}}_n$. Altitude lattices within a family have the same number of linear intervals. They are related to each other via extensions, refinements, and embeddings of some skeletal posets. For a poset $\mathcal{P}$ with $\hat{0}$, we define its Kneser graphs $KG(k):=(V(k),E)$, where $V(k):=\{x:h(x_{\downarrow })=k,1\le k\le h(\mathcal{P})\}$ and $E:=\{(x,y):x_{\downarrow }\cap y_{\downarrow }=\hat{0}\}$. We give some observations about them in a reconstruction setting.

33 pages, 38 figures. Preliminary version of two upcoming works, comments welcome!