  We give a new semi-combinatorial proof for the equality of the number of
ballot permutations of length $n$ and the number of odd order permutations of
length $n$, which is due to Bernardi, Duplantier and Nadeau. Spiro conjectures
that the descent number of ballot permutations and certain cyclic weight of odd
order permutations of the same length are equi-distributed. We present a
bijection to establish a Toeplitz property for ballot permutations with any
fixed number of descents, and a Toeplitz property for odd order permutations
with any fixed cyclic weight. This allows us to refine Spiro's conjecture by
tracking the neighbors of the largest letter in permutations.
