24 problems

For a positive integer $n$, a tuple ${\bf r}=(r_1,\ldots,r_m)$ of nonnegative integers, and the polynomial ${\sf{M}}^n_{\bf r}(q)$ defined in the paper, the claim concerns its coef…

Collins–Dykema–Torres-Ayala conjecture. This coefficient is non-negative.

Let $r,n_1,ots,n_r$ be positive integers, let $[n_i]=\{1,\dots,n_i\}$, and let $q_1,\dots,q_r$ and $x_j^{(i)}$ be variables for $i\in[r]$ and $j\in[n_i]$. For the Cartesian product…

First Borwein conjecture. The polynomials $A_n(q)$, $B_n(q)$ and $C_n(q)$ have nonnegative coefficients.

Negative-coefficient conjecture. In the remaining 33 pairs of the polynomials $A_{abcd}$ and $B_{2-a,2-b,2-c,2-d}$, including the case where $p\neq 2$ for $a=b=c=d=1$, one of the t…

Conditional positivity conjecture. $Q_{01}$ has nonnegative coefficients if and only if $p=2$ and $q\equiv 1 \mod 4$.

Nonnegative-coefficient conjecture. If $r\in\mathbb N$ is arbitrary but fixed, then $h(t,a;r)$, viewed as a polynomial in $(t,a)$, has nonnegative coefficients. This would consider…

Let $G$ be any graph and $0\leq q\leq1$. Let $\mathcal{M}_{ef}(q)$ be the random-cluster polynomial associated with distinct edges $e,f$, and let $A_{\beta,\gamma}$ be the set of…

Let $G$ be a graph with distinct edges $e,f$, and let $\mathcal{M}_{ef}(q)$ be the polynomial defined from the random-cluster partition-function sums. For edge subsets $\beta$ and…

Positivity conjecture for $\mathcal{M}_{ef}(q)$. For $0\leq q\leq1$ and positive edgeweights $x_e$, one has

For positive integers $a,b$ and $k$, define … where $c_d(k)_q=\sum_{s\mid\gcd(d,k)}\mu(d/s)[s]_q$. Here $[n]_q$ is the $q$-integer, $\mu$ is the Möbius function, and…

For each integer $a\geq3$, let $\operatorname{Cat}((a;c))_q$ denote the $q$-Catalan germ associated with an integer $c$, as defined in the paper. q-Catalan germ positivity conjectu…

For coprime positive integers $a$ and $b$, define the rational $q$-Catalan number … Here $[n]_q=(1-q^n)/(1-q)$ and $\mathbb{N}[q]$ denotes polynomials in $q$ with non-negative inte…

Lieb–Seiringer conjecture. This coefficient is non-negative.

Let $A$ and $B$ be $n \times n$ positive semidefinite matrices, and let $m$ and $r$ be integers with $m \geq r \geq 0$. Lieb–Seiringer's conjecture. The coefficient of $t^r$ in … i…

Let $\operatorname{RedVHC}_k(\operatorname{Av}_{3k-i}(312))$ denote the set of $312$-avoiding reduced valid hook configurations with $k$ hooks on permutations with $2k+i$ points. D…

Let $X$ be a hyperkähler manifold, and let … be its Riemann–Roch polynomial. Root and log-concavity conjectures. The following properties are conjectured: 1. The sequence of coeffi…

Neguț's conjecture. If $a_2\geqslant a_3\geqslant \cdots\geqslant a_n\geqslant 0$, then $F(a_2,\ldots,a_n)$ has nonnegative coefficients.

A weak composition is a finite sequence of nonnegative integers. For weak compositions $a$ and $b$, let $\overline{\mathfrak{P}}_a$ denote the kaon associated with $a$, and let…

Let $d$ be a positive integer, let $\ol\nu\vdash d$, and let $s_{\ol\nu}$ be the polynomial map associated with the staircase Kronecker-square coefficients. Let $c(\ol\nu)$ and…

Let $\gamma$ be a clan and let $\Upsilon_{\gamma}(X;Y)$ denote the polynomial associated to the $K$-orbit closure $Y_{\gamma}$. K-orbit multidegree conjecture. Theorem(III) holds f…

Positivity rigidity conjecture. Then $p_0\geq 0$, and for $m=1,2,\ldots,s+1$,

Let $\mathcal{D}^{+}(\mathbb{R})$ denote the class used in the paper, and let $D(f)=\deg_{+}(f)-\deg(f)$ for the relevant polynomial $f$. For a polynomial…

Let $p=(x-\rho)(x-\bar\rho)$ be a quadratic polynomial with nonreal roots, written as $\rho=r(\cos\varphi+\mathbf{i}\sin\varphi)$ with $r>0$ and $\varphi\in(0,\frac{\pi}{2})$, and…