3 problems

Odd-shift inclusion conjecture. If $k$ is odd, then $X_1\subset X_2$.

Let $a$ and $b$ be positive integers with $2a<b$, and let $u$ be a partition that is both $operatorname{Reg}{a,b}$-valid and $operatorname{Colreg}{a,b}$-valid. For each hook…

Let $a$ and $b$ be positive coprime integers with $a<b$, and let $\frac{b}{a}$ be the ratio of these integers. Let $u$ be a partition, and write $H{i,j}$ for the hook at position…