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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.