2 problems

Let $\Sigma$ be a $(d-1)$-dimensional complex admitting a convex-ear decomposition, and let $S\subseteq [d]$. Write $P_\Sigma$ for the face poset of $\Sigma$, and $(P_\Sigma)_S$ fo…

Let $\Pi_n$ be the partition lattice, and let $S$ be a subset of the ranks used for rank selection. Denote by $b_n'(S)$ the multiplicity of the trivial representation of…