6 problems

Let $\mathcal C_n(132,213)$ be the set of cyclic permutations of size $n$ avoiding the patterns $132$ and $213$, and let $\mathcal C_n^B(132,213)$ be the subset of balanced such pe…

For each $n\geq 1$, let $c_n=|\mathcal C_n(321)|$, where $\mathcal C_n(321)$ is the set of cyclic permutations of size $n$ avoiding $321$. The monotonicity conjecture. The sequence…

Minimum-cardinality conjecture. For all $k \geq 1$, the minimum cardinality of an unavoidable subset of $\overline{S_k}$ is

Balanced cyclic composition proportion conjecture. For even $n$,

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…

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