4 problems

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…

Joseph's nonnegativity conjecture. The integers $b_i(n)$ are nonnegative.

Su's conjecture. The equality $b'_S(n)=1$ uniquely characterises the first $r$ consecutive ranks, and $b'_S(n)$ is nonzero for every admissible subset $S$.