5 problems

Let $\mathcal{B}_n$ be the set of ballot permutations of $[n]$, and let $\mathcal{O}_n$ be the set of permutations of $[n]$ whose cycles all have odd length. For distinct letters…

Spiro's conjecture. For every $n\geq1$,

Wang and Zhang's refinement. For all $n$, $d$, and $2\leq j\leq n-1$,

Let $b_{n,d}(i,j)$ denote the number of ballot permutations of length $n$ with descent $d$ and the relevant endpoint data $(i,j)$, and let $p_{n,d}(i,j)$ denote the corresponding n…

Let $n\ge 1$. A ballot permutation is a permutation whose initial segments contain at least as many ascents as descents, and an odd order permutation is a permutation all of whose…