Abstract:
By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual and that it is entirely determined by so-called totally primitive elements (elements such that the two half-coproducts are 0). I will construct a basis indexed by a new combinatorial family called biplane forests in bijection with packed words. In this basis, primitive elements are generated by biplane trees and totally primitive elements by a certain subset of trees.
Thus we obtain the first explicit basis for totally primitive elements of WQSym.