Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra

The Malvenuto–Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto–Reutenauer algebra, both of which include the Loday–Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.