16 problems

For a positive integer $r$, let $b_k(r)=\binom{k+r-1}{r}$, and let $R_n(b_k(r))$ denote the corresponding reciprocal Hankel matrix. Integrality criterion conjecture. The inverse of…

Let $F_n$ be the $n$th Fibonacci number, let $e(n,i,k)$ be the exponent used to determine the sign in the source, and let $d_k(r)$ be the Fibonomial coefficient corresponding to…

Integral congruence conjecture. There is an integer matrix $Y$ such that

Bounded-correction conjecture. For every such 4-tuple,

Villarreal's determinant conjecture. If $\alpha_0(\mathcal C')=\beta_1(\mathcal C')$ for all minors $\mathcal C'$ of $\mathcal C$ and $x^{v_1},\dots,x^{v_q}$ have degree $d\geq 2$,…

Asymptotic counting conjecture. The lower bound gives the true order of growth:

Let $r\geq 2$ and let $A\in \operatorname{Mat}_r(\mathbf Z)$ be nonsingular. Define $\gcd(A-I)$ as the greatest common divisor of the entries of $A-I$, and call $A$ primitive when…

Let $\lambda$ be a partition with $n$ parts, and let $M=(m_{i,j})_{1\le i,j \le n}\in\mathcal{M}_\lambda$ be a minimal matrix of shape $\lambda$. A matrix is Hankel when its entrie…

Let $G$ be the topological ordering of a graph on $[n]$, and let $B$ be the matrix over ${\mathbb{Z}}/p{\mathbb{Z}}$ given by … Let $r_p$ be the rank of $B$ over…

Bounded-excess conjecture. There is a constant $c>0$ such that

Let $R_n(H;f)$ denote the number of matrices $A\in\mathcal M_n(\mathbb Z;H)$ with characteristic polynomial $f\in\mathbb Z[X]$, where $\mathcal M_n(\mathbb Z;H)$ is the set of…

Nilpotence conjecture. For all $a\geqslant 0$, $b\geqslant 0$, and $k\geqslant 0$, there exists an integer $N\geqslant 0$ such that every entry of

Miller–Reiner conjecture. For any differential poset $P$ and any $n\geq 0$, the matrix

Let $\Omega$ be an arbitrary Hessenberg type, and let $NRS(\Omega)$ denote the set of NRS-matrices of type $\Omega$. Call a matrix $\varsigma$-reduced according to the paper's redu…

Uniform error conjecture. For $n\geq 5$ and $\ell\geq 1$, one has

Let $\mathcal C$ be a uniform clutter with edge incidence vectors $v_1,\ldots,v_q$, and suppose its packing property holds. Define … as the matrix whose columns are the augmented v…