11 problems

Dotsenko's embedding conjecture. The operad $\mathrm{FMan}$ is isomorphic to the suboperad of $\mathrm{ComPreLie}$ generated by $x-y$ and $c$.

Let $\mathfrak{L}$ be the index set, let $T$ be the pre-Lie algebra constructed from multi-indices over $\mathfrak{L}$, and let $D$ be its derivation-type pre-Lie product. Write…

Écalle–Chapoton conjecture. The numbers $a_{n,k}$ are given by Sequence A055277, refined by the number of leaves, with parameter $k$.

Écalle–Chapoton conjecture. Any Lie element of ${\bf PBT}$ is a linear combination of L-classes.

Écalle–Chapoton conjecture. The space ${\bf PBT}\cap{\rm Lie}$ is the free pre-Lie algebra generated by ${\bf G}_1={\bf P}_\bullet$.

A pre-Lie commutative algebra has a symmetric associative product $\circ$ and a not-necessarily-symmetric product $\cdot$ satisfying the pre-Lie commutative algebra relations. Ever…

Let $\operatorname{PreLie}$ be the operad of pre-Lie algebras, and let $\operatorname{PreLie}^{[2]}$ denote its second Veronese power. The preceding computations show that all rela…

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 $\mathscr{Y}=\mathscr{E}\circ\mathscr{D}^{-1}$ be the quotient series in the group $\mathsf{G}_{\operatorname{PreLie}}$, let $\mathscr{Y}_T$ be the coefficient of a rooted tree…

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…

Conjectural operad-algebra property. If such an operad ${\mathcal{X}}$ exists, then every cofree-cocommutative right-sided combinatorial Hopf algebra is an ${\mathcal{X}}$-algebra.