6 problems

Sundaram–Welker's acyclicity conjecture. For $\lambda\vdash\mu$ with $\lambda\neq\mu$, the space $X_{\lambda,\mu}$ is ${\mathbb Q}$-acyclic.

Let $\Pi_n$ be the partition lattice, let $S$ be a set of ranks, and let $\Pi_n^S$ denote its rank-selected subposet. Write $\Delta(\Pi_n^S)$ for the order complex of $\Pi_n^S$, le…

Let $\Pi_n$ be the poset of set partitions of $\{1,2,\dots,n\}$ ordered by refinement, and let $S\subseteq\{1,2,\dots,n-2\}$. Write $\beta_S(\Pi_n)$ for the rank-selected homology…

Coefficientwise domination conjecture. The symmetric function $\alpha_{2n}-R_{2n}$ is a nonnegative integer combination of homogeneous symmetric functions $h_\lambda$ indexed by pa…

Su's h-positivity conjecture. The action of $S_{2n}$ on the homology of $\Pi_{2n}^e(\bar{k})$ is a permutation module that can be written as a sum of induced modules whose stabilis…

Let $\Pi_n$ be the partition lattice, let $S\subset\{1,2,\ldots,n-2\}$, and set $i=\max(S)$. Let $\beta_S(\Pi_n)$ denote the rank-selected homology $S_n$-representation. Sharp stab…