10 problems

Let $L(m,n)$ denote the finite Young lattice of partitions that fit inside an $m\times n$ rectangle. A symmetric chain order is a partition of the poset into chains symmetric about…

Let $B_n$ be the Boolean lattice of all subsets of an $n$-element set, ordered by containment, and let $G$ be a subgroup of its automorphism group. The quotient $B_n/G$ consists of…

Let $L(m,n)$ be the minuscule lattice of partitions contained in an $m$ by $n$ box, and let $\#\operatorname{SCD}(L(m,n))$ denote the number of symmetric chain decompositions of th…

For positive integers $m$ and $n$, let $L(m,n)$ be the poset of $m$-tuples … ordered coordinatewise, with rank $\sum_{r=1}^m a_r$. A chain is saturated if it skips no rank, and sym…

Let $L(w,h)$ be the poset of partitions in a box of height $h$ and width $w$, ordered componentwise. A saturated chain in $L(w,h)$ has an edge labelling recording the columns in wh…

A symmetric chain decomposition (SCD) of the hypercube $Q_n$ is a partition of its vertices into symmetric chains, where a symmetric chain is a path $(x_k,x_{k+1},\ldots,x_{n-k})$…

Edge-disjoint SCD conjecture. The $n$-cube has

Symmetric chain order conjecture for $Q_n(d_0,\dots,d_k)$. The poset $Q_n(d_0,\dots,d_k)$ is a symmetric chain order for all $n,d_0,\dots,d_k\geq0$.

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.