10 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…

Let … denote its fundamental bijection (the notation … avoids the vincular patterns … This would describe avoidance in cyclic permutations directly through avoidance in their assoc…

Let $c_n(132)$ denote the number of cyclic permutations of length $n$ that avoid the pattern $132$. Bóna–Cory's conjecture for $132$. The inequality … holds. The procedure develope…

Let $c_n(312)$ denote the number of cyclic permutations of length $n$ that avoid the pattern $312$. Bóna–Cory's conjecture for $312$. The inequality … holds. The paper proves a str…

For a pattern $\sigma\in\mathcal S_3$, let $c_n(\sigma)$ be the number of cyclic permutations of length $n$ that avoid $\sigma$. Bóna–Cory's conjecture. For every…

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…

Reduced-residue circular permutation conjecture. There exist integers $a_1,a_2,\ldots,a_{\varphi(n)}$ such that both $\{a_1,\ldots,a_{\varphi(n)}\}$ and

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