# Documentation

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# `combinat`::`stirling1`

Stirling numbers of the first kind

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## 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 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