6 problems

Positive-definiteness conjecture. Despite the negative signs in the definition of $Q$, the quadratic form $Q$ is positive definite on the cone $C_{\mathbb{R}}$.

Let $m$ and $n$ be coprime positive integers. An $(m,n)$-core is a partition that is simultaneously an $m$-core and an $n$-core; denote the set of such cores by $\mathcal P_{m,n}$.…

Let $m$ and $n$ be coprime positive integers. An $(m,n)$-Dyck path is a lattice path from $(0,0)$ to $(m,n)$ staying on the prescribed side of the main diagonal; let…

Let $\mathsf{Cat}_{a,b}(q,t)$ be the rational $q,t$-Catalan number and let $[m]_q$ and $\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q$ denote the $q$-integer and $q$-binomial coefficient. Rati…

Rational $q,t$-Catalan symmetry conjecture.

Fix $a,b\in\mathbb{N}$ with $\gcd(a,b)=1$. Let $\mathrm{D}(a,b)$ be the set of partitions whose diagrams fit in the triangle with vertices $(0,0)$, $(0,a)$, and $(b,a)$. For a cell…