79 problems

Schinzel's conjecture.

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 $S$ 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 $\boldsymbol{e}_{n,k}$ denote the elementary symmetric Boolean function of degree $k$ in $n$ variables. A Boolean function is balanced when it takes each value equally often, a…

Let $\mathcal N(1,n_0,\dots,n_N)$ be the moduli space of rank-one nested instantons, and let $\mu_0$ range over partitions indexing the summation in the generating function of its…

Let $d$ and $n$ be positive integers, let $X(d,n)$ denote the elementary symmetric polynomial … and let $wt(d)$ denote the number of ones in the binary expansion of $d$. Assume tha…

Let $t$ and $\ell$ be positive integers, and let … be an elementary symmetric polynomial over $GF(2)$ in $2^{t+1}\ell-1$ variables. A polynomial is balanced when it takes each valu…

For a positive integer $k$, let $\mathrm{pre}_k$ be the elementary symmetric partition function, and let $\mathcal{P}_{>k}(n)$ denote the set of partitions of $n$ with length stric…

Let $n$ and $k$ be positive integers, and let $\mathrm{pre}_k$ map a partition with at least $k$ parts to the summands of the elementary symmetric polynomial $e_k$ evaluated at its…

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 $n\not\equiv 2\pmod{3}$.

Hunter's conjecture. The global minimum of $h_{2k}$ on this normalized sphere is attained at $\tilde{a}$.

Let the new class of symmetric multivariable polynomials be the class introduced in the source from traditional one-variable polynomials. Two-relation conjecture. All functions in…

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

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…

Let $n$ be a positive integer, and let $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_\ell)$ be an integer partition of $n$. Define $\operatorname{pre}_2(\boldsymbol{\lambda})$ to…

Fix $n\geq 2$ and $m=2n$. Let $X_{m-1}\subset\mathbb{P}^{m-1}$ be the hypersurface defined by $e_{m-1}(x_1,\dotsc,x_m)=0$, and let $F_{n-1}(X_{m-1})$ denote the Fano scheme of proj…

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…