4 problems

For every admissible multiset parameter $\mathbf{s}$, the associated multiset Eulerian–Narayana polynomial $E_{\mathbf{s}}(t)$ is real-rooted. More precisely, after removing its fa…

For a nonzero real polynomial $r$ of degree $d$, define $\mathcal{W}(r)$ by $\sum_{n\ge 0} r(n)t^n=\frac{\mathcal{W}(r)(t)}{(1-t)^{d+1}}$. If the coefficients of both…

For every pair of integers $1\le k\le n$, consider the refined Narayana polynomial and its formal degree-$k$ symmetric decomposition, written in the form…

Conjecture 2 (Brenti). Let $W$ be a finite Coxeter group. For $u,v\in W$ with $u\leq v$ in Bruhat order, define $\widetilde{R}_{u,v}$ by