7 problems

Let ${\Greekmath 0115}$ be an integer partition, and let $J_{{\Greekmath 0115}}(q)$ be the polynomial associated with the specialization of the monomial symmetric function…

Positivity conjecture. For every partition ${\Greekmath 0115}$, all coefficients $a_k$ of $J_{{\Greekmath 0115}}(q)$ are strictly positive. This is one of three conjectures motivat…

Ismail's characterization conjecture. Then $(P_n)_{n\geq 0}$ are continuous $q$-Jacobi polynomials, Al-Salam–Chihara polynomials, or special or limiting cases of these families. Mo…

Let $c_1,c_2,\ldots,c_k$ be integers with $c_i\geq 2$, and let $M_q(c_1,c_2,\ldots,c_k)$ be the associated matrix. Two sequences are considered equivalent when one is obtained from…

Let $c_1,c_2,\ldots,c_k$ be integers, and let $M_q(c_1,c_2,\ldots,c_k)$ be the associated matrix in $\mathrm{PSL}_q(2,\mathbb{Z})$. Leclere–Morier-Genoud trace unimodality conjectu…

Sagan et al.'s conjecture. The polynomials $A_{n,k}(q)$ are unimodal.

Let $n$ and $k$ be positive integers with $1\leq k<n$, and define … Here $A_{n-1}(q)$ denotes the Eulerian polynomial of degree $n-1$. Coprime-width formula. If $\gcd(k,n)=1$, then…