156 problems

Chapman's conjecture.

For $m,n\in\mathbb{N}$, let $A_n^{(m)}$ be the $n\times n$ matrix defined by … Here $\delta_{ij}$ is the Kronecker delta. Sun's determinant-form conjecture. Sun conjectured that ……

Morales–Pak–Panova's conjecture. Under these assumptions,

Luschny's permanent conjecture. Luschny conjectured that

Jacobi-symbol conjecture for $c_p$. For any odd prime $p$,

Sun's conjecture. The element $2B_p$ is a quadratic residue modulo $p$. This conjecture was later confirmed, so the claim is solved.

Sun's conjectures. The following assertions hold:

Let $r,D\geq 1$ be integers, and let $\Delta_{r,D}$ denote the determinant of the linear system obtained from the Bernoulli-polynomial identities for the coefficients of the restri…

Let $n$ be a positive integer, let $x$ be a variable, and let $a$, $b$, $c$, $d$, and $q$ be parameters. Let $p_n(x;a,b,c,d;q)$ denote the Askey–Wilson polynomial. Quadratic relati…

Let $\rho_n$ denote the maximal determinant of an $n\times n$ $0/1$-matrix. Maximal determinant conjecture. For $n=14$, … At the time of the source, the exact value was reported as…

Bernoulli determinant conjecture. The first determinant equals

Totally symmetric plane-partition determinant conjecture. For every nonnegative integer $n$,

Conjecture 4. Every such row is one of

Let $q$ be a parameter, let $\boldsymbol{V}_{n,q}^{(\beta)}$ be the cyclic $\mathcal{U}_q(\mathfrak{gl}_n)$-module generated by the quantum $\alpha$-determinant with parameter…

Determinant-evaluation conjecture. The following identities should hold:

For positive integers $n$ and $k$, let $\operatorname{STab}((k^n))$ be the set of standard tableaux of rectangular shape $(k^n)$. For each $T\in\operatorname{STab}((k^n))$, let…

Polynomial power-compositions determinant conjecture. For any positive integers $n$ and $k$,

Let $Q_n$ be the random symmetric matrix whose entries take values $1$ and $-1$, as in the stated model. Determinant growth conjecture. Almost surely, … This is the symmetric-matri…

The allowable-path determinant conjecture. For every such Kostant picture,

The determinant conjecture. Every cluster monomial is an MV-polynomial that is naturally expressible as a determinant: if $\psi$ is a cluster monomial, then there is a Kostant pict…

Simion's non-crossing matching conjecture. For every positive integer $n$,

Simion's refined determinant conjecture. For every positive integer $n$,

Simion's type-B determinant conjecture. For every positive integer $n$,

Xin's second determinant conjecture.

Xin's first determinant conjecture. The determinant