9 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.…

Let $\operatorname{PreLie}$ be the free Pre-Lie algebra on one generator, let $\operatorname{Dend}$ be the free Dendriform algebra on one generator, and let $\operatorname{Lie}$ be…

An $L$-dendriform algebra is a vector space equipped with two operations $x\prec y$ and $x\succ y$ satisfying … … where $x * y=x\prec y+x\succ y$. Vallette's conjecture. The operad…

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

The dendriform module on the cyclic group carries an action $\theta$. Let $\mathbf{b}_n$ be the integer sequence used in the even case, and define a second integer sequence…

The dendriform module on the cyclic group carries an action $\theta$. Let $\mathbf{b}_n$ be the integer sequence defined by the coefficients of the symmetric function…