7 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,

Pham's orbit-size growth conjecture. There is an integer $c_P$ such that, for all $k$,

Pham's dual-necklace conjecture. For all primitive necklaces $P$, the rational functions $H_P(x)$ and $H_{P^*}(x)$ can be written with the same denominator of degree $|P|=|P^*|$.

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…

Let $n=T_k+r$ with $k\geq 2$ and $0\leq r\leq k$. Let $D(n,1)$ denote the depth of the subgraph $\mathcal G_{n,1}$ of the Mancala graph induced by the $1$-monotone configurations.…