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Stirling Numbers

Well-known to the reader is probably the following result: If we expand $(x+1)^n$ for a positive integer by the binomial theorem, the occurring coefficients of powers
$$(x+1)^n={n \choose 0}x^n+{n \choose 1}x^{n-1}+{n \choose 2}x^{n-2}+\cdots+{n \choose n-1}x^1+{n \choose n}x^0$$
are called binomial coefficients which obey a recurrence formula $$\binom nk=\binom {n-1}{k-1} + \binom {n-1}{k}.$$ This recurrence formula can be visualized by arranging the binomial coefficients in the so-called Pascal’s triangle.

In this part, we will define Stirling numbers, named after James Stirling (1692 – 1770) that have in many ways properties very similar to those of the binomial coefficients. In particular, it will turn out that there are two types of Stirling numbers, and for both types, there are similar recurrence relationships.

Like for binomial coefficients, Stirling numbers of both types have also interesting combinatorial interpretations.

| | | | created: 2020-04-04 10:21:08 | modified: 2020-04-04 10:21:28 | by: bookofproofs

1.Definition: Stirling Numbers of First and Second Kind

2.Lemma: Stirling Numbers and Rising Factorial Powers

3.Proposition: Recursive Formula for the Stirling Numbers of the First Kind

4.Triangle of the Stirling Numbers of the First Kind

5.Proposition: Recursive Formula for the Stirling Numbers of the Second Kind

6.Triangle of the Stirling Numbers of the Second Kind

7.Explanation: Combinatorial Interpretation of Stirling Numbers of the First Kind

8.Explanation: Combinatorial Interpretation of Stirling Numbers of the Second Kind

9.Proposition: Factorials and Stirling Numbers of the First Kind

10.Proposition: Comparison between the Stirling numbers of the First and Second Kind

11.Proposition: Inversion Formulas For Stirling Numbers

Edit or AddNotationAxiomatic Method

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