General characteristics (1 formula) Series representations (7 formulas) Integral representations (1 formula) Limit representations (1 formula) Generating functions (4 formulas) Identities (4 formulas) Complex characteristics (6 formulas) Summation (23 formulas) Operations (1 formula) Representations through equivalent functions (3 formulas)

Rocky Mountain J. Math. Volume 46, Number 6 (2016), 1919-1923. An explicit formula for Bernoulli polynomials in terms of $\boldsymbol r$-Stirling numbers of the second kind. Bai-Ni Guo, István Mező, and Feng Qi 252 MATHEMATICS MAGAZINE Close Encounters with the Stirling Numbers of the Second Kind KHRISTO N. BOYADZHIEV Ohio Northern University Ada, Ohio 45810 [email protected]

If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact [email protected] for assistance. Stirling numbers of the second kind The Stirling numbers of the second kind describe the number of ways a set with n elements can be partitioned into k disjoint, non-empty subsets. For example, the set {1, 2, 3} can be partitioned into three subsets in the following way --

Let S (n, k) be the Stirling number of the second kind, which is the number of partitions of a set consisting of n elements into k pairwise disjoin t non- empty subsets, and let ˜ Stirling numbers of the second kind The Stirling numbers of the second kind describe the number of ways a set with n elements can be partitioned into k disjoint, non-empty subsets. For example, the set {1, 2, 3} can be partitioned into three subsets in the following way -- Stirling Numbers of the Second Kind Stirling Numbers are named after the Scottish mathematician James Stirling (1692-1770) From a previous page, we showed that some polynomials could be represented as factorials: k=k (1) [1.01] k 2 =k (1) +k (2) [1.02] k 3 =k (1) +3k(2) +k (3) [1.03] We write k on the left hand side rather than x, because ... Stirling Numbers of the Second Kind. The Stirling numbers of the second kind, or Stirling partition numbers, describe the number of ways a set with n elements can be partitioned into k disjoint, non-empty subsets. Common notations are S(n, k), , and , where the first is by far the easiest to type. Can someone help me understand the formula for Stirling numbers of the second kind? When I look at the formula for calculating the Stirling numbers it doesn't really make sense. I like to know the why behind formulas and I'm having trouble figuring this one out.