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Séminaire Lotharingien de Combinatoire, 84B.74 (2020), 12 pp.

# Hugo Mlodecki

# Basis of Totally Primitive Elements of WQSym

**Abstract.**
By Foissy's work, the bidendriform structure of the Word
Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to
its dual. However, the only known explicit isomorphism does not respect the
bidendriform structure. This structure is entirely determined by so-called
totally primitive elements (elements such that the two half-coproducts are 0).
In this paper, we construct a basis indexed by a new combinatorial family
called biplane forests in bijection with packed words. In this basis,
primitive elements are indexed by biplane trees and totally primitive elements
by a certain subset of trees. Thus we obtain the first explicit basis for the
totally primitive elements of WQSym.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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