90 problems

Let $D_1,D_2,D_3$ be positive definite matrices, and let $M$ be the matrix whose entries are the coefficients of $y$ and $z$ in … A minor means the determinant of any finite square…

Let $\mathcal{D}$ be a full-rank cluster algebra of finite type, and consider ratios of products of its cluster variables on the totally positive locus. Such a ratio is called boun…

Hankel total positivity conjecture. The sequences of polynomials

Canonical-basis characterization conjecture.

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…

Let $A$) be a totally nonnegative $n\times n$ matrix, meaning that all its minors are nonnegative, and let $\lambda\vdash n$ be a partition. For a class function $\phi^{\lambda}$ o…

Sokal's functional-digraph conjecture. The following assertions hold: (i) $[\psi_{n,k}]_{n,k}$ is totally positive; (ii) $(\bm{\psi}_n(y))_{n\geq0}$ is coefficientwise Hankel-total…

Let $P_n(x;\bm{\mu})$ be the row-generating polynomials of the Graham–Knuth–Patashnik recurrence, with parameters $\bm{\mu}=(\alpha,\beta,\gamma,\alpha',\beta',\gamma')$ treated as…

Regularity conjecture for tilted Richardson cells. The CW-complex in the preceding stratification is regular.

Let $A$ be a totally positive element of $GL_n$, and let $R$ be a ratio of products of minors as above. In a weighted planar-network parametrization, call $R=p/q$ subtraction free…

Let $G=\operatorname{SL}_n(\mathbb{C})$, let $v=\operatorname{id}$ and $w=w_0$, and let $\twist_v^w$ be the twist automorphism of the open Richardson variety, preserving its totall…

Let $M(m,n)$ be the excedance matrix, and let $N(m,n)$ be obtained by reversing the order of its rows: its $i$th row is the $(m-i+1)$st row of $M(m,n)$. A matrix is totally nonnega…

Let $C^{(r)}$ be the Stirling cycle triangle of order $r$, let … be the triangle formed by reversing the rows of $C^{(r)}$, and let $c_{r,n}(x)$ be its row-generating polynomials.…

Let $M$ be a generic square matrix, and let a double wiring diagram be arbitrary. The chamber-minor Laurent positivity conjecture. Every minor of $M$ can be written as a Laurent po…

Let $m,p>1$, let $\operatorname{Grass}(p,m+p)$ be the Grassmannian, and let $\alpha^1,\ldots,\alpha^n$ be Schubert data. A real upper-triangular matrix with diagonal entries $1$ is…

Let $f(x,y)=\sum f_i(x)y^i\in\mathbf{P}^{\mathrm{pos}}_2$, and form the upper-triangular coefficient matrix … A minor means the determinant of any finite square submatrix. The tota…

Kenyon–Wilson's conjecture. The matrix $M$ is the response matrix of a circular electrical network belonging to a cell of co-dimension $r$ in the poset $EP_n$ if and only if the fi…

Let $\mathcal{F}$ be the barycentric subdivision, and let $H_{\mathcal{F}}$ be the transformation matrix encoding the transformation of the $h$-vector under an $\mathcal{F}$-unifor…

Let $(\mathbf n,\mathbf m)$ be as above, and let $\boldsymbol\alpha_{\mathbf n}$ and $\boldsymbol\beta_{\mathbf m}$ be positive sequences satisfying the stated conditions and summa…

Partial flag Toeplitz limit conjecture. There exists a sequence $u^{(n)}d4d9\text{in}X_{P_{d4c3}^{(n)}}(\mathbb R_{>0})$ converging to $u^{d4c4}$; conversely, every infinite totall…

Fujimoto conjecture. These polynomials are in general position: any $m$ polynomials chosen from them are linearly independent over $\mathbb{C}$. This conjecture is the algebraic in…

Positive-part closure conjecture. The closure of the positive part equals the nonnegative part:

Sokal's conjecture. The Hadamard product $M(\mathbf{x})*M(\mathbf{y})$ is totally monomial positive; equivalently, every minor

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…