The bond lattice $L_G$ of a finite graph G is the collection of set partitions $\pi$ of the vertex set V(G) such that each block of $\pi$ induces a connected subgraph of G. This lattice encodes a gread deal of combinatorial information about G, and in fact is isomorphic to the lattice of flats of G's graphic matroid. The noncrossing bond poset of a graph (first considered systematically by Farmer--Hallam--Smyth in 2020) is the intersection of $L_G$ with the lattice of noncrossing partitions. In general, the structure of the noncrossing bond poset depends heavily on the choice of labeling of G. Indeed, as the name suggests they need not even be lattices.
In this talk I will discuss the surprisingly nice combinatorics that arises when one considers the noncrossing bond poset of a family of graphs called Ferrers graphs. Given a natural choice of vertex labeling, the resulting poset of noncrossing bonds is EL-shellable and its Mobius invariant can be shown to satisfy a "Catalan-like" recurrence. I will also present some conjectures and open problems relating the combinatorics of these posets to lattice walks.