5 problems

Let $T$ be a non-crossing tree of degree $n$, not necessarily based, viewed as an element of the operad of moulds $obreakoperatorname{Mould}$. Let $\operatorname{Dend}$ denote the…

Freeness conjecture. The dendriform trialgebra ${\bf WQSym}$ is free.

Let ${\bf PQSym}^*$ denote the dual Hopf algebra of parking functions, realized as a dendriform trialgebra with operations induced from the dendriform trialgebra structure on…

Let $\mathscr{F}_t$ be the global series introduced from the parameterized flow series, and let $\mathsf{F}_t=-(1-t)\mathbf{N}_t\mathbf{T}\mathbf{P}_t$ be its dendriform candidate.…

Free dendriform trialgebra conjecture. The algebra ${\bf PQSym}^{(l)}$ is a free dendriform trialgebra.