6 problems

Billey–Burdzy–Sagan peak polynomial positivity conjecture. For every admissible set $S$ with $m=\max(S)$, each coefficient $(\Delta^j p_S)(m)$ is a positive integer for…

Let $I\subset[n-1]$ be a peak set, and let $p(n)$ be the peak polynomial, so that the number of permutations in $\mathfrak{S}_n$ with peaks exactly at $I$ is $p(n)2^{|I|-1}$. The d…

Hurwitz stability conjecture. The polynomial $T_{n,r}(x)$ is Hurwitz stable for all $r\ge 2$ and $n\in\mathbb{N}$. The sequence is introduced after Hurwitz stability has been estab…

Let $I$ be an admissible peak set, meaning that $I\subset [n-1]/\{1\}$ and $i\in I$ implies $i+1\notin I$. For $n>\mathrm{max}(I)$, let $P(I,n)$ be the set of permutations in…

Integral and classified real-root conjecture. If all roots of $p_S(x)$ are real, then all of them are integral. Moreover, $p_S(x)$ has only real roots if and only if

Bounded-roots conjecture. Every complex root of $p_S(z)$ lies in