  A permutation whose any prefix has no more descents than ascents is called a
ballot permutation. In this paper, we present a decomposition of ballot
permutations that enables us to construct a bijection between ballot
permutations and odd order permutations, which proves a set-valued extension of
a conjecture due to Spiro using the statistic of peak values. This bijection
also preserves the neighbors of the largest letter in permutations and thus
resolves a refinement of Spiro' s conjecture proposed by Wang and Zhang. Our
decomposition can be extended to well-labelled positive paths, a class of
generalized ballot permutations arising from polytope theory, that were
enumerated by Bernardi, Duplantier and Nadeau. We will also investigate the
enumerative aspect of ballot permutations avoiding a single pattern of length 3
and establish a connection between 213-avoiding ballot permutations and Gessel
walks.
