5 problems

Let $\lambda$ be a partition and let $j$ be an integer in the range for which the symmetric function $Q_{\lambda,j}$ is defined. Representation conjecture. The symmetric function…

Let $E_n(x,q)=\sum_{\sigma\in S_n}x^{\operatorname{des}(\sigma)}q^{w(\sigma)}$ be the $q$-Eulerian polynomial, and fix $k$. Consider the coefficient of $x^k$ in $E_n(x,q)$ as $n$ v…

Shareshian and Wachs's conjecture. For all $r\in[n]$, the palindromic polynomial $A_n^{(r)}(q,t)$ is $q$-unimodal. This is presented as a consequence of the chromatic-quasisymmetri…

For an integer $n\geq 0$, let $B_{n,k}(q)$ denote the type $B$ $q$-Eulerian numbers for $0\leq k\leq n$. A finite sequence is unimodal if there is an index at which it weakly incre…

Positivity conjecture. All the coefficients of the polynomials $G^*_{2n}(q)$ are positive. This would give a coefficientwise positivity refinement of the factorization of the $q$-a…