6 problems

Let $n$ be a positive integer. A cyclic V-shaped permutation of $[n]$ is a permutation with a unique local minimum that is also a cycle, and let the minimum occur at position $k$.…

Let $n,t\geq 2$ and let $w<t$. Let $\mathcal{V}_t(n,w)$ be the universal cycle constructed from the relevant fixed-weight strings, and let $\mathcal{V'}_t(n,w)$ be the sequence obt…

Necklace invertibility conjecture. There exist infinitely many $n$, different from a power of $p$, for which every aperiodic necklace of length $n$ over $\Sigma_p$ with non-zero we…

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 $a$ and $b$ be positive coprime integers. An $(a,b)$-necklace is the combinatorial object considered in the paper, with length denoted by $n$. Asymptotic existence conjecture.…