2 problems

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…

Triangularity conjecture. The matrix of ${{\bf S}^\sigma}^\sharp$ in the ${\bf S}$ basis is triangular. Its diagonal values are $1$ for elements of $X_n$ and $0$ for elements of…