Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0 satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 [(2?b1 a1)k a2 ?(b1 b2)]Tn?1,k ?(a1 ?b1)(k 1)Tn?1,k+1, where initial conditions 0 unless ? k n and T0,0 1. Denote by Tn Pnk =0 Tn,k. In this paper, we show asymptotic normality of give an formula Tn. As applications, many famous combinatorial numbers, such as Stirlin...

P.L. Erdős and L.A. Székely [Adv. Appl. Math. 10(1989), 488– 496] gave a bijection between rooted semilabeled trees and set partitions. L.H. Harper’s results [Ann. Math. Stat. 38(1967), 410–414] on the asymptotic normality of the Stirling numbers of the second kind translates into asymptotic normality of rooted semilabeled trees with given number of vertices, when the number of internal vertice...

For a simple finite graph G denote by { G k } the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If En is the graph on n vertices with no edges then { En k } coincides with { n k } , the ordinary Stirling number of the second kind, and so we refer to { G k } as a graph Stirling number. Harper showed that the ...

Large sample properties of statistical inferences in the context of finite populations are harder to determine than in the iid case due to their dependence jointly on the characteristics of the finite population and the sampling design employed. There have been many discussions on special inference procedures under special sampling designs in the literature. General and comprehensive results ar...