76 problems

Let $A\in\{0,1\}^{n\times n}$ be a matrix with the 2-consecutive ones property, meaning that the ones in each row occur in at most two blocks in some ordering of the columns. 2-con…

Let $T$ be an $m\times n$ table with $2n\geqslant m$. Let $\mathcal{C}(T)$ be the coefficient matrix whose entries are the coefficients arising when the relevant quantities…

Let $R$ be a commutative finite chain ring (CFCR) of nilpotency index $e$, let $n$ be a positive integer, and suppose that the maximal ideal of $R$ is generated by $\gamma$. Write…

Let $R$ be a commutative finite chain ring (CFCR) of nilpotency index $e$, and let $n$ be a positive integer. Let $C_n(R)$ denote the ring of $n\times n$ circulant matrices over…

Let $\rho$ be the evaluation map used in the paper. Assuming the divisibility conjecture, for every integer $n$ define the square matrix $\mathbf{M}_n$ by … Let…

Prove the collection of explicit conjectural evaluations, due to Cigler and to Cigler–Krattenthaler, for shifted Hankel determinants of Catalan-like sequences arising from weighted…

Di Francesco's conjecture.

For positive integers $m,n$ with $n$ odd, define … For primes $p\equiv1\pmod4$, this determinant is an integer square, so choose a square root $\sqrt{D_p^{(3)}}$. Sun's quadratic-c…

Shifted G-determinant identities. For all $n\geq3$, the three chains of identities displayed in the source hold:

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 $G$ be an $R$-disjoint graph. Write $B(G)$ for its bipartite part, and for each odd cycle $C$ of $G$, let $R(C)$ denote the subgraph associated with $C$. Determinantal factoriz…

Determinant-asymptotics conjecture. The leading asymptotics is given by

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…

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…

Xin's second determinant conjecture.

Let $(G,\beta)$ be an edge-labeled graph over a GCD domain $R$, and let $F_1,F_2,\ldots,F_n\in\hat{R}_G$ be splines. The determinant of these splines is denoted by…

The minor-ratio conjecture. If $d_0=f_0=0$ and both omitted sets are finite, then, except for countably many values of $a$,

The Taylor-coefficient limiting ratio conjecture. Except for countably many values of $a$,

The first limiting conjecture. Except for countably many values of $a$,

Johnson's conjecture. For every $n\ge 2$ and every Toeplitz matrix $A\in M_n(\mathbb{R})$ satisfying $A+A^{\top}=2J_n$,

Frozen-corner determinant conjecture. The number $B_{n,s}$ may be evaluated as

For each form $P$, let $\str(P)$ denote the least number of reducible forms whose sum is $P$. Let $\mathrm{det}_n$ be the determinant viewed as a homogeneous polynomial in the rows…