We say that a permutation $\pi$ is ballot if, for all $i$, the word $\pi_1\cdots \pi_i$ has at least as many ascents as it has descents. We say that $\pi$ is an odd order permutation if $\pi$ has odd order in $S_n$. Let $b(n)$ denote the number of ballot permutations of order $n$, and let $p(n)$ denote the number of odd order permutations of order $n$. Callan observed that, seemingly, $b(n)=p(n)$ for all $n$. Whether this is true or not remains unknown. In this paper we conjecture that a stronger statement is true. 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 $d=1,\ 2$, and $d=\lfloor(n-1)/2\rfloor$, and in each of these cases we establish formulas for $b(n,d)$ involving second-order Eulerian numbers and E 查看全文>>