Maximal chains in lattices from graph associahedra: Tamari to the weak order

In this paper, we study the maximal chains of lattices which generalizes both the weak order and the Tamari lattice: certain lattices of maximal tubings. A maximal tubing poset $\mathfrak{L}(G)$ is defined for any graph $G$, but for the graphs we consider in this paper, the poset is a lattice. Just as the weak order is an orientation of the $1$-skeleton of the permutahedron and the Tamari of the associahedron, each tubing lattice is an orientation of the $1$-skeleton of a graph associahedron. The partial order on $\mathfrak{L}(G)$ is given by a projection from $\mathfrak{S}_n$ to $\mathfrak{L}(G)$. In particular, when the graph is the complete graph, the graph associahedron is the the permutahedron, and when it is the path graph, it is the Stasheff associahedron. Our main results are for lollipop graphs, graphs that ``interpolate'' between the path and the complete graphs. For lollipop graphs, the lattices consist of permutations which satisfy a generalization of $312$-avoiding. The maximum length chains correspond to partially shiftable tableaux under the Edelman-Greene's Coxeter-Knuth bijection. We also consider functions defined analogously to Stanley's symmetric function for the maximum length chains and find their expansion in terms of Young quasisymmetric Schur functions.