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# Permutations

Contents:

1. Permuation allocations

2. Methods

   1. Accessing permutation elements

   2. Permuation properties

   3. Permuation functions

   4. Reading and writing permutations

3. Permutations in cyclic form

4. Applying Permutations

## Permuation allocations

- GSL::Permutation.alloc(n)

  These functions create a new permutation of size `n`. The permutation is not initialized and its elements are undefined. Use `GSL::Permutation.calloc` if you want to create a permutation which is initialized to the identity.

- GSL::Permutation.calloc(n)

  This creates a new permutation of size `n` and initializes it to the identity.

## Methods

- GSL::Permutation#init()

  This initializes the permutation to the identity, i.e. (0,1,2,...,n-1).

- GSL::Permutation.memcpy(dest, src)

  This method copies the elements of the permutation `src` into the permutation `dest`. The two permutations must have the same size.

- GSL::Permutation#clone

  This creates a new permutation with the same elements of `self`.

### Accessing permutation elements

- GSL::Permutation#get(i)

  Returns the value of the `i`-th element of the permutation.

- GSL::Permutation#swap(i, j)

  This exchanges the `i`-th and `j`-th elements of the permutation.

### Permutation properties

- GSL::Permutation#size

  Returns the size of the permutation.

- GSL::Permutation#valid

  This checks that the permutation `self` is valid. The n elements should contain each of the numbers 0 .. n-1 once and only once.

- GSL::Permutation#valid?

  This returns true if the permutation `self` is valid, and false otherwise.

### Permutation functions

- GSL::Permutation#reverse

  This reverses the elements of the permutation `self`.

- GSL::Permutation#inverse

  This computes the inverse of the permutation `self`, and returns as a new permutation.

- GSL::Permutation#next

  This method advances the permutation `self` to the next permutation in lexicographic order and returns `GSL::SUCCESS`. If no further permutations are available it returns `GSL::FAILURE` and leaves `self` unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order.

- GSL::Permutation#prev

  This method steps backwards from the permutation `self` to the previous permutation in lexicographic order, returning `GSL_SUCCESS`. If no previous permutation is available it returns `GSL_FAILURE` and leaves `self` unmodified.

### Reading and writing permutations

- GSL::Permutation#fwrite(io)

- GSL::Permutation#fwrite(filename)

- GSL::Permutation#fread(io)

- GSL::Permutation#fread(filename)

- GSL::Permutation#fprintf(io, format = "%un")

- GSL::Permutation#fprintf(filename, format = "%un")

- GSL::Permutation#fscanf(io)

- GSL::Permutation#fscanf(filename)

### Permutations in cyclic Form

A permutation can be represented in both `linear` and `cyclic` notations. The functions described in this section convert between the two forms. The linear notation is an index mapping, and has already been described above. The cyclic notation expresses a permutation as a series of circular rearrangements of groups of elements, or `cycles`.

For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different sets of elements can be combined independently, for example (1 2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange of elements 4 and 5. A cycle of length one represents an element which is unchanged by the permutation and is referred to as a `singleton`.

It can be shown that every permutation can be decomposed into combinations of cycles. The decomposition is not unique, but can always be rearranged into a standard `canonical form` by a reordering of elements. The library uses the canonical form defined in Knuth's `Art of Computer Programming` (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.

The procedure for obtaining the canonical form given by Knuth is,

1. Write all singleton cycles explicitly

2. Within each cycle, put the smallest number first

3. Order the cycles in decreasing order of the first number in the cycle.

For example, the linear representation (2 4 3 0 1) is represented as (1 4) (0 2 3) in canonical form. The permutation corresponds to an exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.

The important property of the canonical form is that it can be reconstructed from the contents of each cycle without the brackets. In addition, by removing the brackets it can be considered as a linear representation of a different permutation. In the example given above the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has many applications in the theory of permutations.

- GSL::Permutation#linear_to_canonical

- GSL::Permutation#to_canonical

  Computes the canonical form of the permutation `self` and returns it as a new `GSL::Permutation`.

- GSL::Permutation#canonical_to_linear

- GSL::Permutation#to_linear

  Converts a permutation `self` in canonical form back into linear form and returns it as a new `GSL::Permutation`.

- GSL::Permutation#inversions

  Counts the number of inversions in the permutation `self`. An inversion is any pair of elements that are not in order. For example, the permutation 2031 has three inversions, corresponding to the pairs (2,0) (2,1) and (3,1). The identity permutation has no inversions.

- GSL::Permutation#linear_cycles

  Counts the number of cycles in the permutation `self`, given in linear form.

- GSL::Permutation#canonical_cycles

  Counts the number of cycles in the permutation `self`, given in canonical form.

## Applying Permutations

- GSL::Permutation::permute(v)

  Applies the permutation `self` to the elements of the vector `v`, considered as a row-vector acted on by a permutation matrix from the right, v' = v P. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation `self` and the vector `v` must have the same length.

- GSL::Permutation::permute_inverse(v)

  Applies the inverse of the permutation `self` to the elements of the vector `v`, considered as a row-vector acted on by an inverse permutation matrix from the right, v' = v P^T. Note that for permutation matrices the inverse is the same as the transpose. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation `self` and the vector `v` must have the same length.

- GSL::Permutation.mul(pa, pb)

  Combines the two permutations `pa` and `pb` into a single permutation `p` and returns it. The permutation `p` is equivalent to applying `pb` first and then `pa`.

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