3 problems

Science Fiction conjecture. The bigraded $\mathfrak{S}_n$-module $\bigcap_{i=1}^{k}V_{\mu^{(i)}}$ has dimension $\frac{n!}{k}$, and its Frobenius characteristic is

Let $S\in\binom{[n]}{k}$ and let $\lambda$ be a partition inside $R(n,k)$. Write $S(\lambda)$ for the collection of $k$-subsets defined by the containment conditions in the paper,…

Let $\lambda$ be an integer partition of $n+1$, and let $\mu^{(1)},\ldots,\mu^{(k)}$ be partitions of $n$, each obtained from $\lambda$ by removing a removable cell from its Young…