10 problems

Eulerian transformation conjecture. The Eulerian transformation preserves log-convexity. The source presents this as a related problem after discussing q-log-convexity of Eulerian…

For $r\geq 2$, let $Q_{\rm C}^{(r)}$ and $Q_{\rm S}^{(r)}$ be the quasi-Eulerian cycle and subset triangles, respectively, and let $M^{\rm rev}$ denote reversal of the rows of a ma…

For $r\geq 1$, let $E^{(r)} = \left(\genfrac{\langle}{\rangle}{0pt}{}{n}{k}^{\! (r)}\right)_{n,k\geq 0}$ be the $r$th-order Eulerian triangle, and let…

Alternating Eulerian–Catalan identity.

Let $\mathrm{FR}_n$ denote the set of Fubini rankings on $n$ competitors, and let $\mathrm{flat\_runs}_k(\mathrm{FR}_n)$ denote the flattened Fubini rankings with $k$ runs of ascen…

Monotone total positivity conjecture. For all $i,j,r\in\mathbb{N}$, the determinant of this submatrix is a monotonically increasing function of $i\in\mathbb{N}$.

Edge-coloring conjecture. The edge-coloring of $G$ determined by the hyperplane types prescribes a choice of connecting sets such that

For $n>1$, let $Q_{n,n-1}(q)$ be the coefficient appearing in the conjectured Hilbert-series expansion of the Gallai algebra $\mathcal G_{n,k}$. Let $E(n-1,j)$ denote the second-or…

Let $\bm{A}=\left(\left\langle\begin{matrix}n\k\end{matrix}\right\rangle^{\rm clean}\right)_{n,k\geq 0}$ be the clean Eulerian triangle, where…

Eulerian-square conjecture. The Eulerian square $A^{\ulcorner}$ is strictly totally positive: all its minors are positive. The conjecture is motivated in the source by the strict t…