17 problems

Let $A = \left(\genfrac{\langle}{\rangle}{0pt}{}{n}{k}\right)_{n,k\geq 0}$ be the Eulerian triangle, whose entries count permutations on $[n]$ by descents. Brenti's conjecture. The…

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

Vertical log-concavity conjecture. For fixed $k \geq 2$, the sequence $\mathrm{BoEl}(n,k)$ for $n \geq k$ is log-concave.

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…

Peterson's generalized Gessel conjecture. The generating polynomials of the $W$-Eulerian numbers $N_{ij}$ are $\gamma$-positive.

Tao Gui's conjecture. There should exist a smooth projective variety $X(W)$ whose Hodge numbers are given by the numbers $N_{ij}$; equivalently, after a rotation of $45$ degrees, t…

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

Eulerian-matrix total positivity conjecture. The matrix $A$ is totally positive.

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 $E_{\{n,k\}}$ denote the Lucas-Eulerian numbers, defined by the Lucas analogue of the Eulerian recurrence. For a polynomial $A_{n,k}(x)$, write $\big\{A_{n,k}\big\}$ for its Lu…

Eulerian-difference conjecture. Fix $n\geq3$. If $n$ is even or $n\in\{3,5,7\}$, then the sequence $(D(n,k))_{k\in\mathbb{N}_0}$ is strictly increasing on…

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…

Unimodality and monotonicity conjecture. Every row in $D$ and in $\widetilde{D}$ is a unimodal sequence. For every $k\geq 1$, the sequences