Ballot Permutations and Odd Order Permutations

A permutation $\pi$ is ballot if, for all $k$, the word ${\pi }_1\cdots {\pi }_k$ has at least as many ascents as it has descents. Let $b(n)$ denote the number of ballot permutations of order $n$, and let $p(n)$ denote the number of permutations which have odd order in the symmetric group $S_n$. Callan conjectured that $b(n)=p(n)$ for all $n$, which was proved by Bernardi, Duplantier, and Nadeau. We propose a refinement of Callan’s original conjecture. Let $b(n,d)$ denote the number of ballot permutations with $d$ descents. Let $p(n,d)$ denote the number of odd order permutations with $M(\pi )=d$, where $M(\pi )$ is a certain statistic related to the cyclic descents of $\pi$. We conjecture that $b(n,d)=p(n,d)$ for all $n$ and $d$. We prove this stronger conjecture for the cases $d=1,2,3$, and $d=\lfloor (n-1)\mathrm{/}2\rfloor$, and in each of these cases we establish formulas for $b(n,d)$ involving Eulerian numbers and Eulerian-Catalan numbers.