19 problems

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

Let $[n]=\{1,2,\ldots,n\}$, and let $\Pi_n$ be the refinement order on the set of partitions of $[n]$. An antichain is a collection of pairwise incomparable elements of this poset,…

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…

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…

Largest-antichain bound. For every $n\in\mathbb{Z}^+$,

Let $L$ be one of the lattices considered above, let ${\overline s}$ be a commitment, and let ${\overline w}$ be a word acting on ${\overline s}$ to produce…

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 $S$.

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…

Minimum-cardinality antichain conjecture. There is a maximal antichain in $\Pi_\kappa$ of cardinality

Minimum-cardinality chain conjecture. It is not possible for $\Pi_{2^\lambda}$ to have a maximal chain of cardinality strictly less than

Let $d>1$, let $n+1$ be divisible by $d$, and let $\Pi_n^d$ and $\Pi_n^{(d)}$ be the divisible partition posets defined in the paper. Write $a(d)$ for the rank parameter used in th…

Let $\Pi_n$ be the partition lattice on $n$ elements, let $\overline{\Pi}_n$ be its proper part, and let $G\subset S_n$ be a subgroup acting on the order complex…

Let $Delta(\overline{\Pi}_n)$ denote the order complex of the proper part of the partition lattice on $n$ elements, and let $G\subset S_n$ be a subgroup acting on it. For $n\geq 3$…

Stanley's symmetric chain decomposition conjecture. The lattice $L(m,n)$ has a symmetric chain decomposition: it can be expressed as a disjoint union of rank-symmetric, saturated c…

Stanley's symmetric chain decomposition conjecture. The lattice $L(m,n)$ can be expressed as a disjoint union of rank-symmetric, saturated chains.