5 problems

Constants-equality conjecture. The algebra of constants is exactly $U^{(1)}$:

Hall-monomial constants conjecture. If $M$ is a Hall monomial that is a constant under the derivation $\delta$, then

Term-count conjecture. For $d\geq L$, the maximal possible number of terms satisfies $Q_{L,d}=Q_{L,L}$ and coincides with the number of terms obtained by expanding $R_L$ in the ave…

Let $F_n$ be the free group of rank $n$, let $\mathcal{L}_n$ be the free Lie algebra associated with the lower central series of $F_n$, and let $\mathrm{Der}^+(\mathcal{L}_n)$ be t…

Pascal descent polynomial conjecture. One has