8 problems

Jack polynomial weak-dominance conjecture. The following statements are equivalent: $\lambda$ weakly dominates $\mu$; for $x\in[1,\infty)^n$,

Jack polynomial dominance conjecture. The following statements are equivalent: $\lambda$ dominates $\mu$; for every $x\in[0,\infty)^n$,

Let $\mathcal{A}$ be a cluster algebra and $\mathcal{C}$ an Artinian monoidal categorification of $\mathcal{A}$. Assume that the simple objects of $\mathcal{C}$ are parametrized by…

Predominant orientation-type conjecture. The partition $\Lambda(n,v,d)$ is predominant among all non-vanishing orientation types of $G$.

Let $A,a,B,b$ be integers with $A \geq B \geq 0$ and $a \geq b \geq 0$. For $m,n \in \mathbb{N}$, let $\mathcal{U}^m_n$ denote the set of compositions of $n$ with $m$ non-negative…

Let $A\in\mathcal{SP}_N$ with associated partition $\phi(A)=\lambda$. For a partition $\mu=(\mu_1\geq\cdots\geq\mu_m>0)$, define … Say that $\mu$ is $\downarrow$-invertible when…

Let $\mathcal{SP}_N$ be the collection of multisets of partitions with total size $N$, let $\phi(A)=\lambda$ be the associated partition, and let $\lambda^{++}$ be defined by … A s…

Let $W_1$ be the first group in an iterated induction construction, and let $\Lambda$ be the $W_1$-graph on … where ${\mathscr{H}}_1$ is of type $A$. For vertices…