A lattice of combinatorial Hopf algebras: Binary trees with multiplicities

Abstract

In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS).