Abstract In this paper, we introduce a kind of combinatorial numbers, D − Stirling numbers, and its special cases. An exponential generating function of the D − Stirling numbers is given. We also present recurrence relations, monotonicity, and limiting properties of the D−Stirling numbers. Applications to statistical probability function estimation and restricted occupancy theory are provided. ...

An interesting 2-adic property of the Stirling numbers of the second kind S(n, k) was conjectured by the author in 1994 and proved by De Wannemacker in 2005: ν2(S(2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n. It was later generalized to ν2(S(c2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n, c ≥ 1 by the author in 2009. Here we provide full and two partial alternative proofs of the generalized version. The proofs are based on...

Let n and k be positive integers, d(k) and ν2(k) denote the number of ones in the binary representation of k and the highest power of two dividing k, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that ν2(S(2, k)) = d(k)−1, 1 ≤ k ≤ 2. Here we prove that ν2(S(c2, k)) = d(k)−1, 1 ≤ k ≤ 2, for any positive integer c. We improve and extend this statement in...

Let the Bessel number of the second kind B(n, k) be the number of set partitions of [n] into k blocks of size one or two, and let the Bessel number of the first kind b(n, k) be the coefficient of x in −yn−1(−x) , where yn(x) is the nth Bessel polynomial. In this paper, we show that Bessel numbers satisfy two properties of Stirling numbers: The two kinds of Bessel numbers are related by inverse ...