Permute 6 different cards (that is, arrange them in a row). How many ways are there? Choose two different cards from a standard deck of 52. How many ways are there? How many subsets of 2 or 3 elements are there for the set containing integers 1 to 10, inclusive? How many 3-digit positive even integers contain exactly one zero? P12,permute the 1st and 2nd rows: M1(1/3),multiply every element of the 1st row by 1/3. A12(-3), add to the elements of the 2nd row, -3 times the corresponding elements of the 1st row: Get more help from Chegg. Get 1:1 help now from expert Advanced Math tutors. List of integers. Permutation pattern, does not include the samples dimension. Indexing starts at 1. For instance, (2, 1) permutes the first and second dimension of the input. Input shape (list of integers, does not include the samples axis) which is required when using this layer as the first layer in a model.
This chapter describes functions for creating and manipulatingpermutations. A permutation is represented by an array of integers in the range 0 to , where each value occurs once and only once. The application of a permutation to a vector yields a new vector where.For example, the array represents a permutationwhich exchanges the last two elements of a four element vector.The corresponding identity permutation is .
Note that the permutations produced by the linear algebra routinescorrespond to the exchange of matrix columns, and so should be consideredas applying to row-vectors in the form rather thancolumn-vectors, when permuting the elements of a vector.
The functions described in this chapter are defined in the header filegsl_permutation.h.
The Permutation struct¶
gsl_permutation¶A permutation is defined by a structure containing two components, the sizeof the permutation and a pointer to the permutation array. The elementsof the permutation array are all of type size_t. Thegsl_permutation structure looks like this:
Permutation allocation¶
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. Usethe function gsl_permutation_calloc() if you want to create apermutation which is initialized to the identity. A null pointer isreturned if insufficient memory is available to create the permutation.
gsl_permutation_calloc(size_t n)¶This function allocates memory for a new permutation of size n andinitializes it to the identity. A null pointer is returned ifinsufficient memory is available to create the permutation.
gsl_permutation_init(gsl_permutation * p)¶This function initializes the permutation p to the identity, i.e.
gsl_permutation_free(gsl_permutation * p)¶This function frees all the memory used by the permutation p.
gsl_permutation_memcpy(gsl_permutation * dest, const gsl_permutation * src)¶This function copies the elements of the permutation src into thepermutation dest. The two permutations must have the same size.
Accessing permutation elements¶
The following functions can be used to access and manipulatepermutations. Loopback 1 2 1 – route audio between applications.
gsl_permutation_get(const gsl_permutation * p, const size_t i)¶This function returns the value of the i-th element of thepermutation p. If i lies outside the allowed range of 0 to then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.
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 thepermutation p.
Permutation properties¶
gsl_permutation_size(const gsl_permutation * p)¶This function returns the size of the permutation p.
gsl_permutation_data(const gsl_permutation * p)¶
This function returns a pointer to the array of elements in thepermutation p.
gsl_permutation_valid(const gsl_permutation * p)¶This function checks that the permutation p is valid. The nelements should contain each of the numbers 0 to n-1 once and onlyonce.
Permutation functions¶
gsl_permutation_reverse(gsl_permutation * p)¶This function reverses the elements of the permutation p.
gsl_permutation_inverse(gsl_permutation * inv, const gsl_permutation * p)¶This function computes the inverse of the permutation p, storingthe result in inv.
Permute 2 2 1 64
gsl_permutation_next(gsl_permutation * p)¶This function advances the permutation p to the next permutationin lexicographic order and returns GSL_SUCCESS. If no furtherpermutations are available it returns GSL_FAILURE and leavesp unmodified. Starting with the identity permutation andrepeatedly applying this function will iterate through all possiblepermutations of a given order.
gsl_permutation_prev(gsl_permutation * p)¶This function steps backwards from the permutation p to theprevious permutation in lexicographic order, returningGSL_SUCCESS. If no previous permutation is available it returnsGSL_FAILURE and leaves p unmodified.
Applying Permutations¶
The following functions are defined in the header files gsl_permute.hand gsl_permute_vector.h.
gsl_permute(const size_t * p, double * data, size_t stride, size_t n)¶This function applies the permutation p to the array data ofsize n with stride stride.
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 thearray data of size n with stride stride.
gsl_permute_vector(const gsl_permutation * p, gsl_vector * v)¶This function applies the permutation p to the elements of thevector v, considered as a row-vector acted on by a permutationmatrix from the right, . The -th column of thepermutation matrix is given by the -th column of theidentity matrix. The permutation p and the vector v musthave the same length.
gsl_permute_vector_inverse(const gsl_permutation * p, gsl_vector * v)¶This function applies the inverse of the permutation p to theelements of the vector v, considered as a row-vector acted on byan inverse permutation matrix from the right, . Notethat for permutation matrices the inverse is the same as the transpose.The -th column of the permutation matrix is given bythe -th column of the identity matrix. The permutation pand the vector v must have the same length.
gsl_permute_matrix(const gsl_permutation * p, gsl_matrix * A)¶This function applies the permutation p to the matrix A fromthe right, . The -th column of thepermutation matrix is given by the -th column of theidentity matrix. This effectively permutes the columns of A accordingto the permutation p, and so the number of columns of A mustequal the size of the permutation p.
gsl_permutation_mul(gsl_permutation * p, const gsl_permutation * pa, const gsl_permutation * pb)¶This function combines the two permutations pa and pb into asingle permutation p, where The permutation p is equivalent to applying pb first andthen pa.
Reading and writing permutations¶
The library provides functions for reading and writing permutations to afile as binary data or formatted text.
gsl_permutation_fwrite(FILE * stream, const gsl_permutation * p)¶This function writes the elements of the permutation p to thestream stream in binary format. The function returnsGSL_EFAILED if there was a problem writing to the file. Since thedata is written in the native binary format it may not be portablebetween different architectures.
gsl_permutation_fread(FILE * stream, gsl_permutation * p)¶This function reads into the permutation p from the open streamstream in binary format. The permutation p must bepreallocated with the correct length since the function uses the size ofp to determine how many bytes to read. The function returnsGSL_EFAILED if there was a problem reading from the file. Thedata is assumed to have been written in the native binary format on thesame architecture.
gsl_permutation_fprintf(FILE * stream, const gsl_permutation * p, const char * format)¶This function writes the elements of the permutation pline-by-line to the stream stream using the format specifierformat, which should be suitable for a type of size_t.In ISO C99 the type modifier z represents size_t, so'%zun' is a suitable format 1.The function returns GSL_EFAILED if there was a problem writingto the file.
gsl_permutation_fscanf(FILE * stream, gsl_permutation * p)¶This function reads formatted data from the stream stream into thepermutation p. The permutation p must be preallocated withthe correct length since the function uses the size of p todetermine how many numbers to read. The function returnsGSL_EFAILED if there was a problem reading from the file.
This function returns a pointer to the array of elements in thepermutation p.
gsl_permutation_valid(const gsl_permutation * p)¶This function checks that the permutation p is valid. The nelements should contain each of the numbers 0 to n-1 once and onlyonce.
Permutation functions¶
gsl_permutation_reverse(gsl_permutation * p)¶This function reverses the elements of the permutation p.
gsl_permutation_inverse(gsl_permutation * inv, const gsl_permutation * p)¶This function computes the inverse of the permutation p, storingthe result in inv.
Permute 2 2 1 64
gsl_permutation_next(gsl_permutation * p)¶This function advances the permutation p to the next permutationin lexicographic order and returns GSL_SUCCESS. If no furtherpermutations are available it returns GSL_FAILURE and leavesp unmodified. Starting with the identity permutation andrepeatedly applying this function will iterate through all possiblepermutations of a given order.
gsl_permutation_prev(gsl_permutation * p)¶This function steps backwards from the permutation p to theprevious permutation in lexicographic order, returningGSL_SUCCESS. If no previous permutation is available it returnsGSL_FAILURE and leaves p unmodified.
Applying Permutations¶
The following functions are defined in the header files gsl_permute.hand gsl_permute_vector.h.
gsl_permute(const size_t * p, double * data, size_t stride, size_t n)¶This function applies the permutation p to the array data ofsize n with stride stride.
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 thearray data of size n with stride stride.
gsl_permute_vector(const gsl_permutation * p, gsl_vector * v)¶This function applies the permutation p to the elements of thevector v, considered as a row-vector acted on by a permutationmatrix from the right, . The -th column of thepermutation matrix is given by the -th column of theidentity matrix. The permutation p and the vector v musthave the same length.
gsl_permute_vector_inverse(const gsl_permutation * p, gsl_vector * v)¶This function applies the inverse of the permutation p to theelements of the vector v, considered as a row-vector acted on byan inverse permutation matrix from the right, . Notethat for permutation matrices the inverse is the same as the transpose.The -th column of the permutation matrix is given bythe -th column of the identity matrix. The permutation pand the vector v must have the same length.
gsl_permute_matrix(const gsl_permutation * p, gsl_matrix * A)¶This function applies the permutation p to the matrix A fromthe right, . The -th column of thepermutation matrix is given by the -th column of theidentity matrix. This effectively permutes the columns of A accordingto the permutation p, and so the number of columns of A mustequal the size of the permutation p.
gsl_permutation_mul(gsl_permutation * p, const gsl_permutation * pa, const gsl_permutation * pb)¶This function combines the two permutations pa and pb into asingle permutation p, where The permutation p is equivalent to applying pb first andthen pa.
Reading and writing permutations¶
The library provides functions for reading and writing permutations to afile as binary data or formatted text.
gsl_permutation_fwrite(FILE * stream, const gsl_permutation * p)¶This function writes the elements of the permutation p to thestream stream in binary format. The function returnsGSL_EFAILED if there was a problem writing to the file. Since thedata is written in the native binary format it may not be portablebetween different architectures.
gsl_permutation_fread(FILE * stream, gsl_permutation * p)¶This function reads into the permutation p from the open streamstream in binary format. The permutation p must bepreallocated with the correct length since the function uses the size ofp to determine how many bytes to read. The function returnsGSL_EFAILED if there was a problem reading from the file. Thedata is assumed to have been written in the native binary format on thesame architecture.
gsl_permutation_fprintf(FILE * stream, const gsl_permutation * p, const char * format)¶This function writes the elements of the permutation pline-by-line to the stream stream using the format specifierformat, which should be suitable for a type of size_t.In ISO C99 the type modifier z represents size_t, so'%zun' is a suitable format 1.The function returns GSL_EFAILED if there was a problem writingto the file.
gsl_permutation_fscanf(FILE * stream, gsl_permutation * p)¶This function reads formatted data from the stream stream into thepermutation p. The permutation p must be preallocated withthe correct length since the function uses the size of p todetermine how many numbers to read. The function returnsGSL_EFAILED if there was a problem reading from the file.
Permutations in cyclic form¶
A permutation can be represented in both linear and cyclicnotations. The functions described in this section convert between thetwo forms. The linear notation is an index mapping, and has alreadybeen described above. The cyclic notation expresses a permutation as aseries of circular rearrangements of groups of elements, orcycles.
Permute 2 2 1 63
For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replacedby 3 and 3 is replaced by 1 in a circular fashion. Cycles of differentsets of elements can be combined independently, for example (1 2 3) (45) combines the cycle (1 2 3) with the cycle (4 5), which is an exchangeof elements 4 and 5. A cycle of length one represents an element whichis unchanged by the permutation and is referred to as a singleton.
It can be shown that every permutation can be decomposed intocombinations of cycles. The decomposition is not unique, but can alwaysbe rearranged into a standard canonical form by a reordering ofelements. The library uses the canonical form defined in Knuth'sArt 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,
Write all singleton cycles explicitly
Within each cycle, put the smallest number first
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 (14) (0 2 3) in canonical form. The permutation corresponds to anexchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
The important property of the canonical form is that it can bereconstructed from the contents of each cycle without the brackets. Inaddition, by removing the brackets it can be considered as a linearrepresentation of a different permutation. In the example given abovethe permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping hasmany applications in the theory of permutations.
gsl_permutation_linear_to_canonical(gsl_permutation * q, const gsl_permutation * p)¶This function computes the canonical form of the permutation p andstores it in the output argument q.
gsl_permutation_canonical_to_linear(gsl_permutation * p, const gsl_permutation * q)¶This function converts a permutation q in canonical form back intolinear form storing it in the output argument p.
gsl_permutation_inversions(const gsl_permutation * p)¶This function counts the number of inversions in the permutationp. An inversion is any pair of elements that are not in order.For example, the permutation 2031 has three inversions, corresponding tothe pairs (2,0) (2,1) and (3,1). The identity permutation has noinversions.
gsl_permutation_linear_cycles(const gsl_permutation * p)¶This function counts the number of cycles in the permutation p Steermouse 5 3 3. , given in linear form.
gsl_permutation_canonical_cycles(const gsl_permutation * q)¶This function counts the number of cycles in the permutation q, given in canonical form.
Examples¶
The example program below creates a random permutation (by shuffling theelements of the identity) and finds its inverse.
Here is the output from the program:
The random permutation p[i] and its inverse q[i] arerelated through the identity p[q[i]]=i, which can be verifiedfrom the output.
The next example program steps forwards through all possible third orderpermutations, starting from the identity,
Here is the output from the program:
The permutations are generated in lexicographic order. To reverse thesequence, begin with the final permutation (which is the reverse of theidentity) and replace gsl_permutation_next() withgsl_permutation_prev().
References and Further Reading¶
The subject of permutations is covered extensively in the following,
Donald E. Knuth, The Art of Computer Programming: Sorting andSearching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
For the definition of the canonical form see,
Donald E. Knuth, The Art of Computer Programming: FundamentalAlgorithms (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.Section 1.3.3, An Unusual Correspondence, p.178–179.
Footnotes
In versions of the GNU C library prior to the ISO C99 standard,the type modifier Z was used instead.
