We introduce probabilistic Stirling numbers of the first kind $$s_Y(n,k)$$s Y ( n , k ) associated with a complex-valued random variable Y satisfying appropriate integrability conditions, thus completing the notion of probabilistic Stirling numbers of the second kind $$S_Y(n,k)$$S Y ( n , k ) previously considered by the first author. Combinatorial interpretations, recursion formulas, and connections between $$s_Y(n,k)$$s Y ( n , k ) and $$S_Y(n,k)$$S Y ( n , k ) are given. We show that such numbers describe a large subset of potential polynomials, on the one hand, and the moments of sums of i. i. d. random variables, on the other, establishing their precise asymptotic behavior without appealing to the central limit theorem. We explicitly compute these numbers when Y has a certain familiar distribution, providing at the same time their combinatorial meaning.

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.

We study a special class of higher order generalized geometric polynomials. Based on our combinatorial interpretation of labeled barred preferential arrangements, we prove several recursions. We al...

The central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other polynomial is related to the anti-diagonal sum of poly-Bernoulli numbers. As an application, we prove Stephan's observation from 2004.