Bubble Lattices II: Combinatorics

TL;DR

This paper introduces two new simplicial complexes related to bubble lattices, explores their combinatorial and geometric properties, proves shellability, and conjectures a connection to the M-triangle of the shuffle lattice.

Contribution

It defines the noncrossing matching and bipartite complexes, proves their shellability, and provides explicit formulas for their face numbers, advancing the understanding of bubble lattice related structures.

Findings

Proved shellability of the complexes.

Derived explicit formulas for refined face numbers.

Conjectured a connection to the M-triangle of the shuffle lattice.

Abstract

We introduce two simplicial complexes, the noncrossing matching complex and the noncrossing bipartite complex. Both complexes are intimately related to the bubble lattice introduced in our earlier article "Bubble Lattices I: Structure" (arXiv:2202.02874). We study these complexes from both an enumerative and a geometric point of view. In particular, we prove that these complexes are shellable and give explicit formulas for certain refined face numbers. Lastly, we conjecture an intriguing connection of these refined face numbers to the so-called M-triangle of the shuffle lattice.