A decomposition of ballot permutations, pattern avoidance and Gessel walks

Abstract

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. c 1 `z 1 ´z has several algebraic or combinatorial interpretations (see [13, A000246]), among which are two classes of restricted permutations:‚ permutations whose any prefix has no more descents than ascents, called ballot permutations; ‚ permutations which are the products of cycles with odd lengths, called odd order permutations. Denote by B n and O n the set of ballot permutations and odd order permutations of length n, respectively. For instance,

Proof of Dilks' bijectivity conjecture on Baxter permutations

“…The integer sequence in (1.2) appears as A005817 in the OEIS [12], where several other intriguing combinatorial interpretations are known. In particular, it has been proved in [11,Theorem 4.8] recently that this sequence also enumerates 231-avoiding ballot permutations. It would be interesting to see whether there is any bijection between these two classes of pattern avoiding permutations.…”