# Permutations

## Introduction

This package defines a `Permutation`

type for Julia. We only
consider permutations of sets of the form `{1,2,3,...,n}`

where `n`

is
a positive integer.

A `Permutation`

object is created from a one-dimensional array of
integers containing each of the values `1`

through `n`

exactly once.

```
julia> a = [4,1,3,2,6,5];
julia> p = Permutation(a)
(1,4,2)(3)(5,6)
```

Observe the `Permutation`

is printed in disjoint cycle format.

The number of elements in a `Permutation`

is determined using the
`length`

function:

```
julia> length(p)
6
```

A `Permutation`

can be converted to an array (equal to the array used
to construct the `Permutation`

in the first place) or can be presented
as a two-row matrix as follows:

```
julia> p.data
6-element Array{Int64,1}:
4
1
3
2
6
5
julia> two_row(p)
2x6 Array{Int64,2}:
1 2 3 4 5 6
4 1 3 2 6 5
```

The evaluation of a `Permutation`

on a particular element is performed
using square bracket or parenthesis notation:

```
julia> p[2]
1
julia> p(2)
1
```

Of course, bad things happen if an inappropriate element is given:

```
julia> p[7]
ERROR: BoundsError()
in getindex at ....
```

To get the cycle structure of a `Permutation`

(not as a character string,
but as an array of arrays), use `cycles`

:

```
julia> cycles(p)
3-element Array{Array{Int64,1},1}:
[1,4,2]
[3]
[5,6]
```

Given a list of disjoint cycles of `1:n`

, we can recover the `Permutation`

:

```
julia> p = RandomPermutation(12)
(1,6,3,4,11,12,7,2,10,8,9,5)
julia> p = RandomPermutation(12)
(1,12,3,9,4,10,2,7)(5,11,8)(6)
julia> c = cycles(p)
3-element Vector{Vector{Int64}}:
[1, 12, 3, 9, 4, 10, 2, 7]
[5, 11, 8]
[6]
julia> Permutation(c)
(1,12,3,9,4,10,2,7)(5,11,8)(6)
```

## Operations

### Composition

Composition is denoted by `*`

:

```
julia> q = Permutation([1,6,2,3,4,5])
(1)(2,6,5,4,3)
julia> p*q
(1,4,3)(2,5)(6)
julia> q*p
(1,3,2)(4,6)(5)
```

Repeated composition is calculated using `^`

, like this: `p^n`

.
The exponent may be negative.

### Inverse

The inverse of a `Permutation`

is computed using `inv`

or as `p'`

:

```
julia> q = inv(p)
(1,2,4)(3)(5,6)
julia> p*q
(1)(2)(3)(4)(5)(6)
```

### Square Root

The `sqrt`

function returns a compositional square root of the permutation.
That is, if `q=sqrt(p)`

then `q*q==p`

. Note that not all permutations have
square roots and square roots are not unique.

```
julia> p
(1,12,7,4)(2,8,3)(5,15,11,14)(6,10,13)(9)
julia> q = sqrt(p)
(1,5,12,15,7,11,4,14)(2,3,8)(6,13,10)(9)
julia> q*q == p
true
```

### Matrix Form

The function `Matrix`

converts a permutation `p`

to a square matrix
whose `i,j`

-entry is `1`

when `i == p[j]`

and `0`

otherwise.

```
julia> p = RandomPermutation(6)
(1,5,2,6)(3)(4)
julia> Matrix(p)
6×6 Array{Int64,2}:
0 0 0 0 0 1
0 0 0 0 1 0
0 0 1 0 0 0
0 0 0 1 0 0
1 0 0 0 0 0
0 1 0 0 0 0
```

Note that a permutation matrix `M`

can be converted back to a `Permutation`

by calling `Permutation(M)`

:

```
julia> p = RandomPermutation(8)
(1,4,5,2,6,8,7)(3)
julia> M = Matrix(p);
julia> q = Permutation(M)
(1,4,5,2,6,8,7)(3)
```

### Sign

The sign of a `Permutation`

is `+1`

for an even permutation and `-1`

for an odd permutation.

```
julia> p = Permutation([2,3,4,1])
(1,2,3,4)
julia> sign(p)
-1
julia> sign(p*p)
1
```

### Reverse

If one thinks of a permutation as a sequence, then applying `reverse`

to that permutation returns a new permutation based on the reversal of
that sequence. Here's an example:

```
julia> p = RandomPermutation(8)
(1,5,8,4,6)(2,3)(7)
julia> two_row(p)
2x8 Array{Int64,2}:
1 2 3 4 5 6 7 8
5 3 2 6 8 1 7 4
julia> two_row(reverse(p))
2x8 Array{Int64,2}:
1 2 3 4 5 6 7 8
4 7 1 8 6 2 3 5
```

## Additional Constructors

For convenience, identity and random permutations can be constructed like this:

```
julia> Permutation(10)
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
julia> RandomPermutation(10)
(1,7,6,10,3,2,8,4)(5,9)
```

In addition, we can use `Permutation(n,k)`

to create the
`k`

'th permutation of the set `{1,2,...,n}`

. Of course,
this requires `k`

to be between `1`

and `n!`

.

```
julia> Permutation(6,701)
(1,6,3)(2,5)(4)
```

The function `Transposition`

is used to create a permutation containing
a single two-cycle. Use `Transposition(n,a,b)`

to create a permutation of
`1:n`

that swaps `a`

and `b`

.

```
julia> p = Transposition(10,3,5)
(1)(2)(3,5)(4)(6)(7)(8)(9)(10)
```

This function requires `1 ≤ a ≠ b ≤ n`

.

## Properties

A *fixed point* of a permutation `p`

is a value `k`

such that
`p[k]==k`

. The function `fixed_points`

returns a list of the fixed
points of a given permutation.

```
julia> p = RandomPermutation(20)
(1,15,10,9,11,13,12,8,5,7,18,6,2)(3)(4,16,17,19)(14)(20)
julia> fixed_points(p)
3-element Array{Int64,1}:
3
14
20
```

The function `longest_increasing`

finds a subsequence of a permutation
whose elements are in increasing order. Likewise, `longest_decreasing`

finds a longest decreasing subsequence.
For example:

```
julia> p = RandomPermutation(10)
(1,3,10)(2)(4)(5,6)(7)(8)(9)
julia> two_row(p)
2x10 Array{Int64,2}:
1 2 3 4 5 6 7 8 9 10
3 2 10 4 6 5 7 8 9 1
julia> longest_increasing(p)
6-element Array{Int64,1}:
3
4
6
7
8
9
julia> longest_decreasing(p)
4-element Array{Int64,1}:
10
6
5
1
```

## Iteration

The function `PermGen`

creates a `Permutation`

iterator.

With an integer argument, `PermGen(n)`

creates an iterator for all permutations of length `n`

.

```
julia> for p in PermGen(3)
println(p)
end
(1)(2)(3)
(1)(2,3)
(1,2)(3)
(1,2,3)
(1,3,2)
(1,3)(2)
```

Alternatively, `PermGen`

may be called with a dictionary of lists or list of lists argument, `d`

.
The permutations generated will have the property that the value of the permutation at argument `k`

must be one of the values stored in `d[k]`

.
For example, to find all derangements of `{1,2,3,4}`

we do this:

```
julia> d = [ [2,3,4], [1,3,4], [1,2,4], [1,2,3] ]
4-element Vector{Vector{Int64}}:
[2, 3, 4]
[1, 3, 4]
[1, 2, 4]
[1, 2, 3]
```

Thus `d[1]`

gives all allowable values for the first position in the permutation, and so forth. We could equally well have done this:

```
julia> d = Dict{Int, Vector{Int}}();
julia> for k=1:4
d[k] = setdiff(1:4,k)
end
julia> d
Dict{Int64, Vector{Int64}} with 4 entries:
4 => [1, 2, 3]
2 => [1, 3, 4]
3 => [1, 2, 4]
1 => [2, 3, 4]
```

In either case, here we create the nine derangements of `{1,2,3,4}`

:

```
julia> [p for p in PermGen(d)]
9-element Vector{Permutation}:
(1,4)(2,3)
(1,3,2,4)
(1,2,3,4)
(1,4,2,3)
(1,3)(2,4)
(1,3,4,2)
(1,2,4,3)
(1,4,3,2)
(1,2)(3,4)
```

The `PermGen`

iterator generates permutations one at a time. So this calculation does not use much memory:

```
julia> sum(length(fixed_points(p)) for p in PermGen(10))
3628800
```

Aside: Notice that the answer is

`10!`

. It is a fun exerice to show that among all the`n!`

permutations of`{1,2,...,n}`

, the number of fixed points is`n!`

.

We provide `DerangeGen(n)`

which generates all derangements of `{1,2,...,n}`

, i.e., all permutations without fixed points.

```
julia> for p in DerangeGen(4)
println(p)
end
(1,2)(3,4)
(1,2,3,4)
(1,2,4,3)
(1,3,4,2)
(1,3)(2,4)
(1,3,2,4)
(1,4,3,2)
(1,4,2,3)
(1,4)(2,3)
```

Thanks to Jonah Scheinerman for the implementation of `PermGen`

for restricted permutations.

`Dict`

Conversion to a For a permutation `p`

, calling `dict(p)`

returns a dictionary that behaves
just like `p`

.

```
julia> p = RandomPermutation(12)
(1,11,6)(2,8,7)(3)(4,5,9,12,10)
julia> d = dict(p)
Dict{Int64,Int64} with 12 entries:
2 => 8
11 => 6
7 => 2
9 => 12
10 => 4
8 => 7
6 => 1
4 => 5
3 => 3
5 => 9
12 => 10
1 => 11
```

## Coxeter Decomposition

Every permutation can be expressed as a product of transpositions. In
a *Coxeter decomposition* the permutation is the product of transpositions
of the form `(j,j+1)`

.
Given a permutation `p`

, we get this form
with `CoxeterDecomposition(p)`

:

```
julia> p = Permutation([2,4,3,5,1,6,8,7])
(1,2,4,5)(3)(6)(7,8)
julia> pp = CoxeterDecomposition(p)
Permutation of 1:8: (1,2)(2,3)(3,4)(2,3)(4,5)(7,8)
```

The original permutation can be recovered like this:

```
julia> Permutation(pp)
(1,2,4,5)(3)(6)(7,8)
```