9 problems

Let ${\operatorname{Sort}}_n$ denote the Coxeter sorting time for permutations of size $n$. Asymptotic sorting-time conjecture. … This conjecture seeks an asymptotically sharp esti…

Fibonacci enumeration conjecture. The sequence $(a_n)$ satisfies

Let $\pi$ be a skew-layered permutation of size $n$, and let $\mathsf{cost}(\pi)$ denote its pop-stack sorting cost. Exclude $\mathsf{id}$ and $-\mathsf{id}$. Skew-layered cost cla…

Consider the step-wise odd-even algorithm depicted in Figure, and let $N$ denote the number of elements being sorted. Its sequential worst-case runtime is compared with Bubble Sort…

Let $P$ be a point configuration on $n^2$ points, and let $f^{(n,\ldots,n)}$ denote the number of standard Young tableaux of the square partition $(n,\ldots,n)$, equivalently the n…

Cycle conjecture. The elements $n$ and $s_n$ are in the same cycle of $\pi$.

Let $(G,c)$ be a bicolored graph, where a press may be performed only at a black vertex, and call a sequence of presses successful if it transforms $(G,c)$ into an all-white, empty…

Holmes–Plummer conjecture. The number of cds-sortable elements in $\mathfrak{S}_{2k+1}$ equals $(k+1)(2k)!$. This conjecture proposes a closed formula for the odd-degree terms in t…

Steingrímsson's sorting conjecture. For all $0\leq t\leq n$, $$