78 problems

Let $\tau$ be a preorder of size $n$ and let $h(\tau,t)$ be its $h$-polynomial. Real-rootedness conjecture. The polynomial $h(\tau,t)$ has only real roots for every preorder $\tau$…

At-most-two non-real zeros conjecture. Every polynomial $W_n(z)$ has at most two non-real zeros.

Self-interlacing conjecture. The polynomial $\Delta_{\theta,h}(p)(x)$ is self-interlacing, or is a self-interlacing polynomial multiplied by $x$.

The Faà di Bruno real-rootedness conjecture. For every $\alpha\in\mathbb{R}$ and $m=0,1,\dots$, $F_m(\alpha,y)$ has only real roots and

Let $\operatorname{GP}(n,k)$ be the generalized Petersen graph, and let $\operatorname{I}(\operatorname{GP}(n,k),x)$ denote its independence polynomial. For all integers…

Brenti's conjecture. For any labeled poset $(P,\omega)$, the polynomial $W(P,\omega;t)$ has only real zeros as a polynomial in $t$. This is the Poset Conjecture, also called the Ne…

Let $P$ be a simplicial poset, with associated polynomials $\operatorname{H}_{\widehat{P}}(x)$, $\operatorname{H}_{{\widehat{P}}^\ast}(x)$, and augmented polynomial…

Let $\mathrm{M}$ be a matroid, let $\mathcal{L}(\mathrm{M})$ be its lattice of flats, and let $\Delta(\widehat{\mathcal{L}}(\mathrm{M}))$ be the order complex of its proper nonempt…

Dilks–Petersen–Stembridge conjecture. For any irreducible finite Weyl group $W$, the affine Eulerian polynomial $\widetilde{W}(x)$ has only real zeros. The conjecture is an affine…

Stahl's real-rootedness conjecture. For every graph $G$, the genus polynomial $\Gamma_G(x)$ has only real roots.

Generalized Laguerre conjecture. Given $t>0$, the polynomial $h_n(x;t)$ satisfies

Let $P$ be a $d$-dimensional lattice polytope, and let $h^*(rP)$ denote the $h^*$-polynomial of its dilation $rP$. Beck–Stapledon conjecture. The polynomial $h^*(rP)$ has only dist…

Let $C^{(r)}$ be the Stirling cycle triangle of order $r$, let … be the triangle formed by reversing the rows of $C^{(r)}$, and let $c_{r,n}(x)$ be its row-generating polynomials.…

Let $W_{n,k}(t)$ be the polynomials considered in the paper. A finite sequence of real-rooted polynomials is Sturm-unimodal if it increases by interlacing up to one index and then…

Bóna's real-rootedness conjecture. The descent polynomial $W_{n,t}(x)$ has only real zeros for any integer $1 \le t \le n-1$.

Let $A=(a_{ij})$ be a real $n\times n$ matrix with non-negative entries, and let $J$ denote the all-ones matrix. Assume that the entries of $A$ are weakly increasing down columns.…

Johnson's conjecture. If $A$ is positive semidefinite, then the polynomial $\eta(zA,-B)$ has all real zeros.

For each graph family listed in Example, let its genus polynomial be the polynomial whose coefficients count embeddings by genus. Stahl's conjecture. The zeros of the genus polynom…

Neggers-Stanley conjecture. For any labeled poset $(P,\omega)$, the polynomial $W(P,\omega;t)$ has only real zeros.

Eventual real-rootedness conjecture. For all sufficiently large $n$, every zero of $q_n^{u,v}$ is real.

Let $A$ and $D$ denote the Eulerian and Delannoy triangles, respectively, and let $A^2$ and $D^2$ be their matrix squares. For a lower triangular matrix $M$, write its $n$-th row g…

Let $\tau$ be a preorder and let $\mathcal Q_\tau$ be its preorder polytope. Magic-positivity conjecture. The Ehrhart polynomial $\operatorname{Ehr}(\mathcal Q_\tau,t)$ is magic po…

Let $\sigma$ be an arbor, and let $\mathcal Q_\sigma$ be its arbor polytope. Cha's real-rootedness conjecture. All roots of the Ehrhart polynomial of $\mathcal Q_\sigma$ are real a…

Let $B_n$ be the type B polynomials defined in the paper, for $n\geq 1$. A polynomial is real-rooted if all of its roots are real. Type B real-rootedness conjecture. For all…

Flag nested-set real-rootedness conjecture. If $\mathbf{N}(\mathcal{L},\mathcal{G})$ is flag, then its $h$-polynomial is real-rooted.