6 problems

Let $\mathbf{m}\in\mathbb{E}_n$, let $\lambda\vdash n$, and let $T\in\operatorname{HikSYT}(\mathbf{m},\lambda)$. For the recursively defined Hikita weight $h_T(q)$ and every real…

Let $a$ and $b$ be integers with $0\leq k\leq a<b$. The codicil conjecture. The following equivalent-looking coefficientwise inequalities hold: … or … Here $\binom{n}{k}_q$ denotes…

Let $a$ and $b$ be integers with $0<a<b$, and let $\beta\geq1$. The $\beta$-conjecture. One has … where the inequality is coefficientwise for polynomials in $q$. This is the three-…

Zanello's conjecture. Its coefficients are non-negative and unimodal. The source describes this as a strengthening of the preceding Gaussian inequality and notes that Zanello treat…

Let $a,b,c,d$ be positive integers satisfying … The Gaussian polynomial $\binom{n}{k}_q$ is the $q$-analogue of the binomial coefficient, and $f(q)\geq g(q)$ means that $f(q)-g(q)$…

Let $M_{i,i+1}^{(\ell)}$ be the update matrix … Normalize $b_r^{(\ell)}=\begin{bmatrix}\ell\\r\end{bmatrix}\widetilde b_r^{(\ell)}$, where … Because the update matrix is symmetric…