Real Zeros and Normal Distribution for Statistics on Stirling Permutations Defined by Gessel and Stanley

Abstract

We study Stirling permutations defined by Gessel and Stanley in [6]. 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.

Generalized Stirling permutations, families of increasing trees and urn models

“…For k = 2, this yields the univariate limit theorems by Bóna [6] and the multivariate limit theorem by Janson [17] for X n , Y n , Z n .…”

Large deviations for the leaves in some random trees

“…This ends the proof if the degree of p n+1 is m + k 0 , which occurs when s n = m. If s n = m then the degree of p n+1 is m + 1 + k 0 , and to end the proof, we want to show that the last (m + k 0 + 1)th root of p n+1 (u) is located to the left of u 1 , so that all negative roots of p n+1 (u) must be simple. To see this, we again follow [10]. The first equation in (3.1) shows that p n+1 (u 1 ) and p n (u 1 ) have opposite signs, and that p n (u 1 ) = 0 as u 1 is a simple root.…”

On the free Lie algebra with multiple brackets

“…Note that the maximal ascending adjacent sequences of θ are (246, 37, 1, 5, 8,9) which are also the maximal terminally nested sequences of ξ(θ). These observations hold in general.…”

“…If θ = 233772499468861551, the map c : [9] → P defined by the pairs (i, c(i)): 1), (2, 3), (3, 3), (4, 2), (5, 2), (6, 1), (7, 1), (8, 2), (9, 1)} is an AA coloring, but {(1, 1), (2,2), (3,3), (4, 3), (5, 2), (6, 1), (7, 1), (8,2), (9, 1)} is not since 24 is an adjacent ascending pair but c(2) = 2 < 3 = c(4).…”