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

A decomposition of ballot permutations, pattern avoidance and Gessel walks