7 problems

Let $M_n$ be the maximum length of a directed cycle of permutations in $\mathcal S_n$ satisfying the Kendall tau distance constraint. For odd $n\geq 5$, Horowitz and Etzion proved…

Let $\mathcal S_n$ be the symmetric group, let $d$ be the Kendall tau metric, and let $M_n$ be the maximum length of a directed cycle of permutations in $\mathcal S_n$ such that no…

Super-long-snake conjecture. There exists a $(2n+1,M_{2n+1},\mathcal{K})$-snake with

Longer-snake conjecture. There exists a $(2n+1,M_{2n+1},\mathcal{K})$-snake with

Horovitz–Etzion's spanning-tree conjecture. The desired spanning tree always exists for odd lengths $2n+1\ge5$.

Let $S_{2n+1}$ be the symmetric group, and let $C$ be a $(2n+1,M,\mathcal{K})$-snake, where $M$ is its number of permutations. Optimal-size conjecture. The optimal size is … The pa…

Let $S_n$ denote the symmetric group on $n$ elements. The direct construction produces a snake-in-the-box code on $S_7$ and $S_9$. Direct-construction conjecture. The construction…