A combinatorial model for the canonical join complex of alt $\nu$-Tamari lattices

TL;DR

This paper introduces a combinatorial model for the canonical join complex of alt ν-Tamari lattices, enabling structural analysis such as vertex decomposability, shelling order, and homology.

Contribution

The paper presents a new combinatorial model that systematically studies the canonical join complex of alt ν-Tamari lattices, revealing their topological properties.

Findings

Proves vertex decomposability of the canonical join complex

Establishes an explicit shelling order

Reveals the homology of the complex

Abstract

Alt $\nu$-Tamari lattices constitute a remarkable family of lattices associated with lattice paths that broadly generalize the Dyck and Tamari lattices. To systematically study the structural properties of this family, we introduce a combinatorial model that realizes the canonical join complex of alt $\nu$-Tamari lattices. Serving as a universal tool, this model allows us to prove vertex decomposability, establish an explicit shelling order, and reveal the underlying homology of the canonical join complex of alt $\nu$-Tamari lattices.