Real Zeros and Normal Distribution for statistics on Stirling permutations defined by Gessel and Stanley

TL;DR

This paper proves that the generating function for descents in Gessel and Stanley's Stirling permutations has only real roots, and uses this to show that the distribution of descents converges to a normal distribution.

Contribution

It establishes the real-rootedness of the generating function and demonstrates normal convergence for descent statistics in these Stirling permutations.

Findings

Generating function has only real roots.

Descent distribution converges to a normal distribution.

Applicable to other equidistributed statistics.

Abstract

We study Stirling permutations defined by Gessel and Stanley. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents, and other, equidistributed statistics on these objects converge to a normal distribution.