Refined Eulerian numbers and ballot permutations

Tongyuan Zhao, Yue Sun, Feng Zhao

Abstract: In this article, we obtained a formula in close form for the multivariate generating function of {A(n,d,j)}n,d,j<math><msub is="true"><mrow is="true"><mo stretchy="false" is="true">{</mo><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">d</mi><mo is="true">,</mo><mi is="true">j</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">}</mo></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">,</mo><mi is="true">d</mi><mo is="true">,</mo><mi is="true">j</mi></mrow></msub></math>, where A(n,d,j)<math><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">d</mi><mo is="true">,</mo><mi is="true">j</mi><mo stretchy="false" is="true">)</mo></math> denote the number of permutations of length n with d descents and j as the first letter. In addition to this, by a series of calculations with generatingfunctionology, we confirm a recent enumerative conjecture of Wang and Zhang for ballot permutations, which are permutations in any prefix of which the descent number is not more than the ascent number.