The peak and descent statistics over ballot permutations

Abstract

A ballot permutation is a permutation π such that in any prefix of π the descent number is not more than the ascent number. By using a reversalconcatenation map, we give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, and connect this distribution and the joint distribution (pk, dp, des) of the peak, depth, and descent statistics over ordinary permutations in terms of generating functions. As corollaries, we obtain several formulas for the bivariate generating function for (i) the peak statistic over ballot permutations, (ii) the descent statistic over ballot permutations, and (iii) the depth statistic over ordinary permutations. In particular, we confirm Spiros conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.

Refined Eulerian numbers and ballot permutations

“…In this subsection we will give a relation between B(t, x, y), B(t, x) and E(t, x, y) by the relations between their coefficients. We give a bijection proof adopting the idea of the reversal-concatenation map in [32]. The concept of lowest point in [32] play a important role in the proof, which is defined to be the position 1…”

“…We give a bijection proof adopting the idea of the reversal-concatenation map in [32]. The concept of lowest point in [32] play a important role in the proof, which is defined to be the position 1…”

“…In order to give a refinement for Theorem 1.1, Spiro [1] introduced a statistic M(π) which is defined for a permutation π such that where |c| is the length of c. Spiro conjectured that the number of ballot permutations of length n with d descents equals the number of odd order permutations π of length n such that M(π) = d, which is confirmed by Wang and the first author [32], and thus Theorem 1.2 follows.…”

“…Wang and the first author [32] give the following Theorem by calculating the joint distribution of the peak and descent statistics over ballot permutations.…”

A decomposition of ballot permutations, pattern avoidance and Gessel walks

“…For k P rns and π P S n , ‚ if 1 ă π ´1pkq " ℓ ă n, then the letters πpℓ ´1q and πpℓ `1q are called the neighbors of k in π; ‚ if πpkq ‰ k, then the letters π ´1pkq and πpkq are called the cyclic neighbors of k in π. Motivated by Gessel's combinatorial interpretation of a decomposition of formal Laurent series in terms of lattice paths [7], Wang and Zhao [21] found a reversal-concatenation decomposition of ballot permutations and proved Conjecture 1.1. Afterwards, Sun and Zhao [19] completed a proof of Conjecture 1.2.…”