"Concrete Mathematics"--a special number

Whether in number theory or combinatorial mathematics, there are some special series-Fibonacci number, Euler number, Stirling number, Cattleya, this article, the author will lead the reader to explore how the past generations of mathematicians from some simple basic problems to extract these special series.

Sterling Number:

There are two types of Stirling, based on the problem of the different situations, we first introduce the second kind of Stirling number.

The second class of Stirling numbers is based on a problem model: dividing the set of n elements into K non-empty sets (denoted by S (n,k)), how many cases are there?

First we start with a few simple examples, obviously, for any n>0,k=1, there s (n,k) = S (n,n) = 1.

And when n = 0 o'clock? S (0,1) =? We use words to describe what this expression wants to say, the number of cases in which the empty set is divided into a non-empty collection, apparently nonexistent, i.e. s (0,1) = 0.

Let's take a closer look at the case of k = 2, for S (n,2), we can consider it as adding the nth element to an arbitrary subset of the first n-1 elements of the collection, since this choice ultimately does not form an empty set, it is necessary to exclude the case that s (n,2) = 2^ (n-1)-1.

This paper analyzes the two special values of K, we try to use the method of analysis k=2 to promote the analysis. Consider S (n,k), which we can consider as a series of operations on a set of n-1 elements. For the nth element, there are two cases:

1. The nth element is a separate group, in which case s (n-1,k-1) occurs.

2. The nth element is not a separate group, and a k*s (n-1,k) situation occurs.

Together, the recursive formula S (n,k) = S (n-1,k-1) + k*s (n-1,k) of the second class of Stirling numbers are obtained.

"Concrete Mathematics"--a special number