It is shown that the sequence of the generalized Bell polynomials S_n(x) is convex under some restrictions of the parameters involved. A kind of recurrence relation for S_n(x) is established, and some numbers related to the generalized Bell numbers and their properties are investigated.

Ther-Whitney numbers of the second kind are a generalization of all the Stirling-type numbers of the second kind which are in line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas forr-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.

The r-Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for r-Bell numbers with real arguments is obtained.

A barred preferential arrangement is a preferential arrangement, onto which in-between the blocks of the preferential arrangement a number of identical bars are inserted. We offer a generalisation of barred preferential arrangements by making use of the generalised Stirling numbers proposed by Hsu and Shiue (1998). We discuss how these generalised barred preferential arrangements offer a unified combinatorial interpretation of geometric polynomials. We also discuss asymptotic properties of these numbers. (C) 2019 The Authors. Published by Elsevier B.V.