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This chapter describes functions for creating and manipulating permutations. A permutation p is represented by an array of n integers in the range 0 .. n-1, where each value p_i occurs once and only once. The application of a permutation p to a vector v yields a new vector v' where v'_i = v_{p_i}. For example, the array (0,1,3,2) represents a permutation which exchanges the last two elements of a four element vector. The corresponding identity permutation is (0,1,2,3).

Note that the permutations produced by the linear algebra routines correspond to the exchange of matrix columns, and so should be considered as applying to row-vectors in the form v' = v P rather than column-vectors, when permuting the elements of a vector.

The functions described in this chapter are defined in the header file
``gsl_permutation.h'`.

A permutation is stored by a structure containing two components, the size
of the permutation and a pointer to the permutation array. The elements
of the permutation array are all of type `size_t`

. The
`gsl_permutation`

structure looks like this,

typedef struct { size_t size; size_t * data; } gsl_permutation;

__Function:__gsl_permutation ***gsl_permutation_alloc***(size_t*`n`)-
This function allocates memory for a new permutation of size
`n`. The permutation is not initialized and its elements are undefined. Use the function`gsl_permutation_calloc`

if you want to create a permutation which is initialized to the identity. A null pointer is returned if insufficient memory is available to create the permutation.

__Function:__gsl_permutation ***gsl_permutation_calloc***(size_t*`n`)-
This function allocates memory for a new permutation of size
`n`and initializes it to the identity. A null pointer is returned if insufficient memory is available to create the permutation.

__Function:__void**gsl_permutation_init***(gsl_permutation **`p`)-
This function initializes the permutation
`p`to the identity, i.e. (0,1,2,...,n-1).

__Function:__void**gsl_permutation_free***(gsl_permutation **`p`)-
This function frees all the memory used by the permutation
`p`.

The following functions can be used to access and manipulate permutations.

__Function:__size_t**gsl_permutation_get***(const gsl_permutation **`p`, const size_t`i`)-
This function returns the value of the
`i`-th element of the permutation`p`. If`i`lies outside the allowed range of 0 to`n`-1 then the error handler is invoked and 0 is returned.

__Function:__int**gsl_permutation_swap***(gsl_permutation **`p`, const size_t`i`, const size_t`j`)-
This function exchanges the
`i`-th and`j`-th elements of the permutation`p`.

__Function:__size_t**gsl_permutation_size***(const gsl_permutation **`p`)-
This function returns the size of the permutation
`p`.

__Function:__size_t ***gsl_permutation_data***(const gsl_permutation **`p`)-
This function returns a pointer to the array of elements in the
permutation
`p`.

__Function:__int**gsl_permutation_valid***(gsl_permutation **`p`)-
This function checks that the permutation
`p`is valid. The`n`elements should contain each of the numbers 0 ..`n`-1 once and only once.

__Function:__void**gsl_permutation_reverse***(gsl_permutation **`p`)-
This function reverses the elements of the permutation
`p`.

__Function:__int**gsl_permutation_inverse***(gsl_permutation **`inv`, const gsl_permutation *`p`)-
This function computes the inverse of the permutation
`p`, storing the result in`inv`.

__Function:__int**gsl_permutation_next***(gsl_permutation **`p`)-
This function advances the permutation
`p`to the next permutation in lexicographic order and returns`GSL_SUCCESS`

. If no further permutations are available it returns`GSL_FAILURE`

and leaves`p`unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order.

__Function:__int**gsl_permutation_prev***(gsl_permutation **`p`)-
This function steps backwards from the permutation
`p`to the previous permutation in lexicographic order, returning`GSL_SUCCESS`

. If no previous permutation is available it returns`GSL_FAILURE`

and leaves`p`unmodified.

__Function:__int**gsl_permute***(const size_t **`p`, double *`data`, size_t`stride`, size_t`n`)-
This function applies the permutation
`p`to the array`data`of size`n`with stride`stride`.

__Function:__int**gsl_permute_inverse***(const size_t **`p`, double *`data`, size_t`stride`, size_t`n`)-
This function applies the inverse of the permutation
`p`to the array`data`of size`n`with stride`stride`.

__Function:__int**gsl_permute_vector***(const gsl_permutation **`p`, gsl_vector *`v`)-
This function applies the permutation
`p`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`p`and the vector`v`must have the same length.

__Function:__int**gsl_permute_vector_inverse***(const gsl_permutation **`p`, gsl_vector *`v`)-
This function applies the inverse of the permutation
`p`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`p`and the vector`v`must have the same length.

The library provides functions for reading and writing permutations to a file as binary data or formatted text.

__Function:__int**gsl_permutation_fwrite***(FILE **`stream`, const gsl_permutation *`p`)-
This function writes the elements of the permutation
`p`to the stream`stream`in binary format. The function returns`GSL_EFAILED`

if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

__Function:__int**gsl_permutation_fread***(FILE **`stream`, gsl_permutation *`p`)-
This function reads into the permutation
`p`from the open stream`stream`in binary format. The permutation`p`must be preallocated with the correct length since the function uses the size of`p`to determine how many bytes to read. The function returns`GSL_EFAILED`

if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

__Function:__int**gsl_permutation_fprintf***(FILE **`stream`, const gsl_permutation *`p`, const char *`format`)-
This function writes the elements of the permutation
`p`line-by-line to the stream`stream`using the format specifier`format`, which should be suitable for a type of`size_t`. On a GNU system the type modifier`Z`

represents`size_t`

, so`"%Zu\n"`

is a suitable format. The function returns`GSL_EFAILED`

if there was a problem writing to the file.

__Function:__int**gsl_permutation_fscanf***(FILE **`stream`, gsl_permutation *`p`)-
This function reads formatted data from the stream
`stream`into the permutation`p`. The permutation`p`must be preallocated with the correct length since the function uses the size of`p`to determine how many numbers to read. The function returns`GSL_EFAILED`

if there was a problem reading from the file.

The example program below creates a random permutation by shuffling and finds its inverse.

#include <stdio.h> #include <gsl/gsl_rng.h> #include <gsl/gsl_randist.h> #include <gsl/gsl_permutation.h> int main (void) { const size_t N = 10; const gsl_rng_type * T; gsl_rng * r; gsl_permutation * p = gsl_permutation_alloc (N); gsl_permutation * q = gsl_permutation_alloc (N); gsl_rng_env_setup(); T = gsl_rng_default; r = gsl_rng_alloc (T); printf("initial permutation:"); gsl_permutation_init (p); gsl_permutation_fprintf (stdout, p, " %u"); printf("\n"); printf(" random permutation:"); gsl_ran_shuffle (r, p->data, N, sizeof(size_t)); gsl_permutation_fprintf (stdout, p, " %u"); printf("\n"); printf("inverse permutation:"); gsl_permutation_invert (q, p); gsl_permutation_fprintf (stdout, q, " %u"); printf("\n"); return 0; }

Here is the output from the program,

bash$ ./a.out initial permutation: 0 1 2 3 4 5 6 7 8 9 random permutation: 1 3 5 2 7 6 0 4 9 8 inverse permutation: 6 0 3 1 7 2 5 4 9 8

The random permutation `p[i]`

and its inverse `q[i]`

are
related through the identity `p[q[i]] = i`

, which can be verified
from the output.

The next example program steps forwards through all possible 3-rd order permutations, starting from the identity,

#include <stdio.h> #include <gsl/gsl_permutation.h> int main (void) { gsl_permutation * p = gsl_permutation_alloc (3); gsl_permutation_init (p); do { gsl_permutation_fprintf (stdout, p, " %u"); printf("\n"); } while (gsl_permutation_next(p) == GSL_SUCCESS); return 0; }

Here is the output from the program,

bash$ ./a.out 0 1 2 0 2 1 1 0 2 1 2 0 2 0 1 2 1 0

All 6 permutations are generated in lexicographic order. To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace `gsl_permutation_next`

with
`gsl_permutation_prev`

.

The subject of permutations is covered extensively in Knuth's Sorting and Searching,

- Donald E. Knuth, The Art of Computer Programming: Sorting and Searching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

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