53 problems

Let ${\bf x}^m=\{x_1,\ldots,x_m\}$ and ${\bf y}^m=\{y_1,\ldots,y_m\}$. For an arbitrary homogeneous symmetric polynomial $S_n({\bf x}^m)$ of degree $n$, write … where…

Let ${\bf x}^m=\{x_1,\ldots,x_m\}$ and let ${\bf 1}^m$ denote the $m$-tuple all of whose entries are $1$. Let $U_n({\bf x}^m,{\bf y}^m)$ be the function defined from the symmetric…

Generalized Bell-polynomial relation. The relation

Conjecture. The only values of $n\in\mathbb{N}_{\geq 3}$ such that $M(n)=2$ are

Let $S$ be the numerical semigroup under consideration, with multiplicity $m$, generators $d_1,\dots,d_m$, gap power sums $G_r(S)$, and invariants $K_p(S)$. Write…

Let $\mathbb{F}_q$ be a finite field, let $3\leq k\leq q-2$ (or $4\leq k\leq q-3$ if $q$ is even), and let $h$ be a complete symmetric polynomial of positive degree $m$ in $k$ vari…

Let $\mathbb{F}_q$ be a finite field, let $3\leq k\leq q-2$ (or $4\leq k\leq q-3$ if $q$ is even), and let $h_e(x_1,\ldots,x_k)$ denote the complete symmetric polynomial of degree…

Betti-number conjecture. The ideals $I_h$ and $I_e$ have the same Betti numbers if and only if $n=3a$ for some integer $a$.

Let $n$ be a positive integer. A partition of $n$ is a weakly decreasing sequence of positive integers with size $n$ and length $\ell$. For $1\leq j\leq\ell$, define…

Let $\lambda$ be a partition, with length denoted by $\ell$, and let $\operatorname{pre}_j(\lambda)$ be the partition whose parts are all products of $j$ distinct parts of…

Terminal stability conjecture. For each fixed $r\ge2$ there exists a finite list of square multigraphs $G_1,\ldots,G_N$, depending only on $r$ and not on $n$, such that for all…

Square-graph cone conjecture. For every $n\ge2$ and every $0\le k\le n-2$, the polynomial $\mathcal D_k^{(n)}$ belongs to the square-graph cone. Equivalently, $\mathcal D_k^{(n)}$…

Modified Ballantine–Beck–Merca conjecture. For $k\geq 3$, $pre_k$ is injective on partitions of $n$ with $j$ parts for each $j\geq k+1$.

Three-variable Schur maximal-rank conjecture. For all sufficiently large $d$, the following holds: if $\lambda\in F$, then $s_\lambda$ fails to be a maximal rank element on $A$; if…

Maximal-rank conjecture for $h_d$. The element $h_d$ is a maximal rank element on $A$ if and only if either $d=2$, $d=3$ and $n\not\equiv 1\pmod{3}$, or $d=4$ and $n=3$.

Maximal-rank conjecture for $e_d$. The element $e_d$ is a maximal rank element on $A$ if and only if either $d=2$, or $d=3$ and $n\not\equiv 2\pmod{3}$.

Schinzel's conjecture.

Let $\vec{k}$ be a vector of positive integers, let $\mathcal{K}$ be the associated parameter, and define the area-bounce polynomial … Write $\lambda(\mathcal{K})$ for the partitio…

Alternant divisibility and positivity conjecture. For every $\alpha\in\mathbb{N}^N$, the alternant $a_\alpha$ is a multiple of $a_\rho$ in $\mathcal{P}$. Furthermore, if…

For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$, let $\operatorname{pre}_2(\lambda)$ be the partition whose parts are the products $\lambda_i\lambda_k$ for…

For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$, let $\operatorname{pre}_i(\lambda)$ be the partition whose parts are all products $\lambda_{i_1}\cdots\lambda_{i_i}$ with…

Let $\vec{c}=(c_1,\dots,c_n)=(1^L2^M3^N)$ and suppose that $q_{\vec{c}}\neq1$. Let $f^{\vec{c}}_{11}$ be the corresponding elliptic corner-VOA structure function and…

Let $m,n\in\mathbb{Z}^{\geq 1}$ with $m\leq n$, and let $\lambda\in\operatorname{Par}(m)$. Let $\widetilde{\psi}^{(q,\xi)}_{\lambda}$ be the specialization map that sends a symmetr…

Let $\Gamma_{n,d}$ be the class of symmetric degree-$d$ polynomials in $n$ variables. For a symmetric hyperbolic polynomial $p$, let its associated operator be the operator defined…

Let $\Gamma_{n,d}$ denote the class of symmetric degree-$d$ polynomials in $n$ variables, and let $\Delta_{\vec{1},\vec{1}}p$ be the indicated second directional derivative. A poly…