\section{Conclusion}

Our next obvious step is to adapt the decomposition through maximum
of~\cref{Phi,tau} to the dual in order to get an explicit bidendriform
isomorphism from $\WQSym$ to its dual. Another generalization which should be
easy is to do the same for the Hopf algebra $\PQSym$ of parking
functions. Indeed $\PQSym$ is a bidendriform bialgebra~\cite{novthi_2006}, and
thus self-dual, but no explicit isomorphism with the dual is known.


\acknowledgments{
The computation and tests needed for this research were done using the
open-source mathematical software \textsc{Sage} and its combinatorics
features developed by the \textsc{Sage-combinat} community. I
particularly thank F. Hivert and V. Pons for all comments on the
writing.
}


