6 problems

Let $\mu\vdash m$ and $\nu\vdash n$ be integer partitions with no common part. Let $\pi$ and $\sigma$ be permutations of cycle types $\mu$ and $\nu$, respectively, with disjoint su…

For each $n$, let $[n-1]=\{1,\ldots,n-1\}$, and let $\beta_n(I)$ denote the number of permutations in $S_n$ with descent set $I\subseteq[n-1]$, while $\beta^{\mathrm{cyc}}_n(I)$ de…

Inverse descent-class commutation conjecture. For every fine set ${\mathcal B}\subset {\mathcal S}_n$ and every $J\subseteq [n-1]$,

Cycle–inverse descent-class conjecture. For every $J\subseteq [n-2]$,

Eulerian descent-set distribution conjecture. For any $n$ and any subset $D\subseteq\{1,2,\dots,n-1\}$,

Eli's descent-set conjecture. For any $n$ and any $I\subseteq[n-1]$,