Refined Eulerian numbers and ballot permutations

TL;DR

This paper derives a closed-form formula for the multivariate generating function of ballot permutations, confirming a recent conjecture and providing new insights into their combinatorial structure.

Contribution

It introduces a closed-form formula for the generating function of ballot permutations and verifies a recent conjecture in the field.

Findings

Derived a closed-form formula for the generating function of ballot permutations.

Confirmed Wang and Zhang's conjecture regarding ballot permutations.

Provided new combinatorial insights into permutations with descent and ascent constraints.

Abstract

A ballot permutation is a permutation {\pi} such that in any prefix of {\pi} the descent number is not more than the ascent number. In this article, we obtained a formula in close form for the multivariate generating function of {A(n,d,j)}, which denote the number of permutations of length n with d descents and j as the first letter. Besides, by a series of calculations with generatingfunctionology, we confirm a recent conjecture of Wang and Zhang for ballot permutations.