4 problems

Equal-constant denominator conjecture. For any two primitive necklaces $P_1$ and $P_2$, if $c_{P_1}=c_{P_2}$, then $H_{P_1}(x)$ and $H_{P_2}(x)$ have the same denominator.

Pham's duality conjecture. For any primitive necklace $P$ such that $c_P$ and $c_{P^*}$ both exist,

Let $P$ be a primitive necklace, and let $\mathcal{O}_{P^k}$ denote the Bulgarian Solitaire orbit parametrized by $P^k$. For a necklace $P'$, say that $P$ and $P'$ are complementar…

Let a sequence of well-behaved $csigma_n$-solitaires have stable configurations with limit shape $cphi$. Limit-shape conjecture. If $cphi$ is a limit shape of the stable configurat…