7 problems

Realization conjecture. There exists a tower of algebras $A_n$, with a basis $(e_\sigma)_{\sigma\in\mathfrak S_n}$, such that the restriction of $\rho_{m,n}$ to canonical words is…

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 ${\bf G}_1={\bf P}_\bullet$.

Let $s$ and $t$ be planar binary rooted trees of the same arity $n$. Let $X=\{0,1,2,\infty\}$ carry the commutative magma operation whose values on $\{0,1,2\}$ are determined by…

A planar binary rooted tree has leaves, internal edges, and a root. A YY-graph is obtained by splicing two planar binary rooted trees $s$ and $t$ of the same arity along their corr…

Freeness conjecture. The algebra ${\bf PBT}^\sharp$ is free over the set ${\bf P}_T^\sharp$, where $T$ runs over trees with at least two nodes and such that the right subtree of th…