118 problems

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 $P=\{P_n(x)\}_{n\geq 0}$ be a polynomial sequence with $\deg(P_n(x))=n$, and write … The associated Eulerian transformation is the linear transformation obtained by taking…

Athanasiadis–Kalampogia-Evangelinou conjecture. For every matroid $\mathsf{M}$, $h_{\mathsf{M}}(x)$ has only nonpositive real zeros.

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

Ferroni's real-rootedness conjecture. Both $\operatorname{\underline{H}}_M(x)$ and $\operatorname{H}_M(x)$ should be real-rooted.

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

Let $P$ be a lattice polytope in $\mathbb{R}^d$, and let $h^*_{rP}(z)$ denote its unweighted Ehrhart $h^*$-polynomial after dilation by the integer $r$. Beck–Stapledon conjecture.…

Neggers' conjecture. These $(P,\omega)$-Eulerian polynomials have only real roots.

For $n,r\in\mathbb{N}$, let $\mathfrak{S}_n^r$ be the set of $r$-stack-sortable permutations and let $A(\mathfrak{S}_n^r;x)$ denote its descent polynomial. Bona's conjecture. The p…

Let $\mu$ be a rigid tile and let $d$ be its associated positive integer. For each $n\geq 1$, let $P_{\mu,d,n}(t)$ denote the tiling polynomial, and write $f(t)\ll g(t)$ when $f$ i…

Let $(\mathrm{M},\mathcal{G})$ be a flag built matroid, and let $\mathbf{N}(\mathrm{M},\mathcal{G})$ be its nested set complex. Nested-set real-rootedness conjecture. The $h$-polyn…

Xie–Zhang's conjecture. For every matroid $\mathsf{M}$, the polynomial $\mathscr{B}(Q_{\mathsf{M}}(t))$ is real-rooted. Real-rootedness would imply strong coefficient inequalities…

Let $\tau$ be an arbor and let $\mathcal Q_\tau$ be its associated arbor polytope. Let ${\rm Ehr}(\mathcal Q_\tau,t)$ denote its Ehrhart polynomial, and let…

Let $\mathrm{M}$ be a matroid, and let its Chow polynomial be the polynomial invariant associated with its lattice of flats. Ferroni–Schröter's conjecture. The Chow polynomial of…

Let $f$ be a real entire function, meaning an entire function real-valued on the real axis, and suppose that both $f$ and $f”$ have only real zeros. Let $LP$ denote the Laguerre–Pó…

Brenti–Welker's conjecture. The chain polynomial of the face lattice of every polytope is real-rooted.

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

Let $p$ and $q$ be integers with $p<q$, let ${\mathbf b}=(b_1,\ldots,b_q)$ satisfy $b_i>0$, and let ${\mathbf a}=(a_1,\ldots,a_p)$ satisfy $a_i\geq b_i$ for $1\leq i\leq p$. The ge…

Negative-real-root conjecture. For any matroid $M$, all roots of $P_M(t)$ lie on the negative real axis.

A genus polynomial is the polynomial whose coefficients record the genus distribution of a graph. Stahl's conjecture. Every genus polynomial is real-rooted. This conjecture was dis…

For a sequence $\{a_k\}_{k\geq0}$, define the operator $\mathcal{L}$ by … A Pólya frequency (PF) sequence is a sequence whose associated generating polynomial has the relevant tota…

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 $f(x,y)\in\mathbf{P}_2$ and $g\in\mathbf{P}$. A polynomial satisfies derivative-substitution when all its derivatives with respect to $y$, evaluated at every real $y$-value, ar…

Let $f\in\mathbf{P}^{\mathrm{pos}}$ and define the Hankel matrix from its derivatives by … For a fixed size $d$, write $\det H((f(x),f'(x),f^{(2)}(x),\dots);d)$ for the correspondi…

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.…