  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 $d=\lfloor(n-1)/2\rfloor$, and in each of these cases
we establish formulas for $b(n,d)$ involving Eulerian numbers and
Eulerian-Catalan numbers.
