Maximal length chains in lattices from graph associahedra

Tubing posets are orientations of the 1-skeleton of graph
associahedra. For the complete graph on $n$ vertices, the poset is the
weak order lattice on the symmetric group. When the graph is the path
on n vertices, the poset is the Tamari lattice. We consider tubing
posets of a family of graphs which interpolates between the complete
and the path graphs. We discuss enumeration of the maximal length
chains of the lattices corresponding to this graph family, first in
terms of standard Young tableaux, then using quasisymmetric functions.
We assume no prior knowledge of any of the topics in the talk. This is
joint work with Samantha Dahlberg.