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# combinat::stirling1

Stirling numbers of the first kind

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```combinat::stirling1(n, k)
```

## Description

combinat::stirling1(n,k) computes the Stirling numbers of the first kind.

Let S(n, k) be the number of permutations of n symbols that have exactly k cycles. Then combinat::stirling1(n,k) computes (- 1)(n + k)S(n, k).

Let S1(n, k) be the Stirling number of the first kind, then we have:

.

## Examples

### Example 1

Let us have a look what's the result of x (x - 1) (x - 2) (x - 3) (x - 4) (x - 5) written as a sum.

`expand(x*(x-1)*(x-2)*(x-3)*(x-4)*(x-5))`

Now let us "prove" the formula mentioned in the "Details" section by calculating the proper Stirling numbers:

```combinat::stirling1(6,1);
combinat::stirling1(6,2);
combinat::stirling1(6,3);
combinat::stirling1(6,4);
combinat::stirling1(6,5);
combinat::stirling1(6,6)```

### Example 2

`combinat::stirling1(3,-1)`
```Error: Nonnegative integers are expected. [combinat::stirling1]
```

## Parameters

 n, k Nonnegative integers

Integer.

## References

J.J. Rotman, An Introduction to the Theory of Groups, 3rd Edition, Wm. C. Brown Publishers, Dubuque, 1988