Represents a permutation of {0,1,...,n-1}. More...

#include <maths/perm.h>

Public Types
using	Index = typename IntOfMinSize<(imageBits *n+7)/8 >::type
	Denotes a native signed integer type large enough to count all permutations on n elements. More...

using	ImagePack = typename IntOfMinSize<(imageBits *n+7)/8 >::utype
	Indicates the native unsigned integer type used to store a single image pack. More...

using	Code = ImagePack
	Indicates the native unsigned integer type used to store the internal permutation code. More...

Public Member Functions
constexpr	Perm ()
	Creates the identity permutation. More...

constexpr	Perm (int a, int b)
	Creates the transposition of a and b. More...

constexpr	Perm (const std::array< int, n > &image)
	Creates a permutation mapping i to image[i] for each 0 ≤ i < n. More...

constexpr	Perm (const Perm &)=default
	Creates a permutation that is a clone of the given permutation. More...

constexpr Code	permCode () const
	Returns the internal code representing this permutation. More...

void	setPermCode (Code code)
	Sets this permutation to that represented by the given internal code. More...

constexpr ImagePack	imagePack () const
	Returns the image pack that represents this permutation. More...

Perm &	operator= (const Perm &cloneMe)=default
	Sets this permutation to be equal to the given permutation. More...

constexpr Perm	operator* (const Perm &q) const
	Returns the composition of this permutation with the given permutation. More...

Perm	cachedComp (const Perm &q) const
	An alias for the composition operator, provided to assist with writing generic code. More...

Perm	cachedComp (const Perm &q, const Perm &r) const
	Deprecated alias for using the composition operator twice, provided to assist with writing generic code. More...

constexpr Perm	conjugate (const Perm &q) const
	Computes the conjugate of this permutation by q. More...

Perm	cachedConjugate (const Perm &q) const
	An alias for conjugate(), provided to assist with writing generic code. More...

constexpr Perm	inverse () const
	Finds the inverse of this permutation. More...

Perm	cachedInverse () const
	Finds the inverse of this permutation, optimised using precomputed "partial lookup tables". More...

constexpr Perm	pow (long exp) const
	Computes the given power of this permutation. More...

Perm	cachedPow (long exp) const
	An alias for pow(), provided to assist with writing generic code. More...

constexpr int	order () const
	Returns the order of this permutation. More...

int	cachedOrder () const
	An alias for order(), provided to assist with writing generic code. More...

constexpr Perm	reverse () const
	Finds the reverse of this permutation. More...

constexpr int	sign () const
	Determines the sign of this permutation. More...

constexpr int	operator[] (int source) const
	Determines the image of the given integer under this permutation. More...

constexpr int	pre (int image) const
	Determines the preimage of the given integer under this permutation. More...

constexpr bool	operator== (const Perm &other) const
	Determines if this is equal to the given permutation. More...

constexpr bool	operator!= (const Perm &other) const
	Determines if this differs from the given permutation. More...

constexpr int	compareWith (const Perm &other) const
	Lexicographically compares the images of (0,1,...,n-1) under this and the given permutation. More...

constexpr bool	isIdentity () const
	Determines if this is the identity permutation. More...

Perm &	operator++ ()
	A preincrement operator that changes this to be the next permutation in the array Perm<n>::Sn. More...

Perm	operator++ (int)
	A postincrement operator that changes this to be the next permutation in the array Perm<n>::Sn. More...

constexpr bool	operator< (const Perm &rhs) const
	Determines if this appears earlier than the given permutation in the array Perm<n>::Sn. More...

constexpr Index	SnIndex () const
	Returns the index of this permutation in the Perm<n>::Sn array. More...

constexpr Index	orderedSnIndex () const
	Returns the lexicographical index of this permutation. More...

std::string	str () const
	Returns a string representation of this permutation. More...

std::string	trunc (int len) const
	Returns a prefix of the string representation of this permutation, containing only the images of the first len integers. More...

void	tightEncode (std::ostream &out) const
	Writes the tight encoding of this permutation to the given output stream. More...

std::string	tightEncoding () const
	Returns the tight encoding of this permutation. More...

void	clear (unsigned from)
	Resets the images of all integers from from onwards to the identity map. More...

constexpr bool	isConjugacyMinimal () const
	Is this permutation minimal in its conjugacy class? More...

Static Public Member Functions
static void	precompute ()
	Performs the precomputation necessary for using the optimised cachedInverse() routine. More...

static constexpr Perm	fromPermCode (Code code)
	Creates a permutation from the given internal code. More...

static constexpr bool	isPermCode (Code code)
	Determines whether the given integer is a valid internal permutation code. More...

static constexpr Perm	fromImagePack (ImagePack pack)
	Creates a permutation from the given image pack. More...

static constexpr bool	isImagePack (ImagePack pack)
	Determines whether the given argument is the image pack of some n-element permutation. More...

static constexpr Perm	rot (int i)
	Returns the ith rotation. More...

static Perm	rand (bool even=false)
	Returns a random permutation on n elements. More...

template<class URBG >
static Perm	rand (URBG &&gen, bool even=false)
	Returns a random permutation on n elements, using the given uniform random bit generator. More...

static Perm	tightDecoding (const std::string &enc)
	Reconstructs a permutation from its given tight encoding. More...

static Perm	tightDecode (std::istream &input)
	Reconstructs a permutation from its given tight encoding. More...

template<int k>
static constexpr Perm	extend (Perm< k > p)
	Extends a k-element permutation to an n-element permutation, where 2 ≤ k < n. More...

template<int k>
static constexpr Perm	contract (Perm< k > p)
	Restricts a k-element permutation to an n-element permutation, where k > n. More...

Static Public Attributes
static constexpr int	imageBits = regina::bitsRequired(n)
	Indicates the number of bits used in an image pack to store the image of a single integer. More...

static constexpr ImagePack	imageMask
	A bitmask whose lowest imageBits bits are 1, and whose remaining higher order bits are all 0. More...

static constexpr PermCodeType	codeType = PERM_CODE_IMAGES
	Indicates what type of internal permutation code is used by this instance of the Perm class template. More...

static constexpr Index	nPerms = factorial(n)
	The total number of permutations on n elements. More...

static constexpr Index	nPerms_1 = factorial(n-1)
	The total number of permutations on n-1 elements. More...

static constexpr SnLookup	Sn {}
	Gives array-like access to all possible permutations of n elements. More...

static constexpr OrderedSnLookup	orderedSn {}
	Gives array-like access to all possible permutations of n elements in lexicographical order. More...

Protected Member Functions
constexpr	Perm (Code code)
	Creates a permutation from the given internal code. More...

Protected Attributes
Code	code_
	The internal code representing this permutation. More...

Detailed Description

template<int n>
class regina::Perm< n >

Represents a permutation of {0,1,...,n-1}.

Amongst other things, such permutations are used to describe simplex gluings in (n-1)-manifold triangulations.

Perm objects are small enough to pass by value and swap with std::swap(), with no need to use references, specialised move operations or custom swap functions. The trade-off is that, for this to be possible, the Perm template class can only work with n ≤ 16.

Each permutation has an internal code, which is a single native integer that is sufficient to reconstruct the entire permutation. Thus the internal code may be a useful means for passing permutation objects to and from the engine. These codes are constructed as follows:

For 8 ≤ n ≤ 16, the code is an image pack: essentially a packed array that holds the images of 0,...,n-1 in a single native integer type. More precisely, this is an unsigned integer of type ImagePack, whose lowest imageBits bits represent the image of 0, whose next lowest imageBits bits represent the image of 1, and so on. This scheme is consistent with the old first-generation codes for n = 4,...,7, which are still supported but no longer used internally.
For n ≤ 7, the code is an index into a hard-coded list of all possible permutations; more precisely, an index into the symmetric group Perm<n>::Sn. The ordering of Perm<n>::Sn is "almost lexicographic", in that we swap some pairs of indices (2k, 2k+1) to ensure that the even permutations are precisely those with even indices.

For n = 2,...,5 (which appear throughout 2-, 3- and 4-manifold triangulations), this template is specialised: the code is highly optimised and also offers some extra functionality. For n = 6,7, this template is again specialised and highly optimised, and it offers some extra functionality but not as much as Perm<5> and below. For n ≥ 8, this template is generic and most operations require more time (in particular, there are no harded-coded lookup tables).

Python: Python does not support templates. For each n = 2,...,16, this class is available in Python under the corresponding name Perm2, Perm3, ..., Perm16.

Template Parameters

n	the number of objects being permuted. This must be between 2 and 16 inclusive.

Member Typedef Documentation

◆ Code

template<int n>

using regina::Perm< n >::Code = ImagePack

Indicates the native unsigned integer type used to store the internal permutation code.

This type alias is present for most values of n, though its precise size depends on how the permutation code is constructed. However, for n = 4,...,7 it is replaced by two type aliases Code1 and Code2, which describe the older first-generation and newer second-generation permutation codes respectively.

◆ ImagePack

template<int n>

using regina::Perm< n >::ImagePack = typename IntOfMinSize<(imageBits * n + 7) / 8>::utype

Indicates the native unsigned integer type used to store a single image pack.

See the class notes for more information on image packs, and how they are used as permutation codes for n ≥ 8.

◆ Index

template<int n>

using regina::Perm< n >::Index = typename IntOfMinSize<(imageBits * n + 7) / 8>::type

Denotes a native signed integer type large enough to count all permutations on n elements.

In other words, this is a native signed integer type large enough to store (n!).

Note: This type is not hyper-optimised: for very small n where Index is hard-coded this is just an int, and for larger n where Index is derived it is actually large enough to hold an entire image pack. If at any stage there are plans to optimise this type, be careful of the special case of n = 8, where (8!) can be stored in an unsigned 16-bit type but not a signed 16-bit type.

Constructor & Destructor Documentation

◆ Perm() [1/5]

template<int n>

constexpr regina::Perm< n >::Perm

inlineconstexpr

Creates the identity permutation.

◆ Perm() [2/5]

template<int n>

constexpr regina::Perm< n >::Perm	(	int	a,
		int	b
	)

inlineconstexpr

Creates the transposition of a and b.

Note that a and b need not be distinct.

Precondition: 0 ≤ a,b < n.

Parameters

a	the element to switch with b.
b	the element to switch with a.

◆ Perm() [3/5]

template<int n>

constexpr regina::Perm< n >::Perm ( const std::array< int, n > & image )

inlineconstexpr

Creates a permutation mapping i to image[i] for each 0 ≤ i < n.

Precondition: The elements of image are 0,...,n-1 in some order.

Parameters

image the array of images.

◆ Perm() [4/5]

template<int n>

constexpr regina::Perm< n >::Perm ( const Perm< n > & )

constexprdefault

Creates a permutation that is a clone of the given permutation.

◆ Perm() [5/5]

template<int n>

constexpr regina::Perm< n >::Perm ( Code code )

inlineconstexprprotected

Creates a permutation from the given internal code.

Precondition: the given code is a valid permutation code; see isPermCode() for details.

Parameters

code	the internal code from which the new permutation will be created.

Member Function Documentation

◆ cachedComp() [1/2]

template<int n>

Perm< n > regina::Perm< n >::cachedComp ( const Perm< n > & q ) const

inline

An alias for the composition operator, provided to assist with writing generic code.

This generic Perm<n> class does not use precomputation to compute compositions. The only point of having cachedComp() in this generic Perm<n> class is to make it easier to write generic code that works with Perm<n> for any n.

If you know you are only working with the generic Perm<n>, you should just use the composition operator instead.
If you are writing generic code, you must remember to call precompute() at least once in the lifetime of this program before using cachedComp().

The permutation that is returned is the same as you would obtain by calling (*this) * q.

Precondition: You must have called precompute() at least once in the lifetime of this program before calling cachedComp(). For generic Perm<n>, precompute() does not affect compositions; however, for other Perm<n> classes a failure to do this will almost certainly crash your program.

Parameters

q	the permutation to compose this with.

Returns: the composition of both permutations.

◆ cachedComp() [2/2]

template<int n>

Perm< n > regina::Perm< n >::cachedComp	(	const Perm< n > &	q,
		const Perm< n > &	r
	)		const

inline

Deprecated alias for using the composition operator twice, provided to assist with writing generic code.

The permutation that is returned is the same as you would obtain by calling (*this) * q * r.

Deprecated:: The three-way cachedComp() was originally written to support conjugation. If you are indeed conjugating, then call cachedConjugate() instead; otherwise just call the two-way cachedComp() twice.

Precondition: You must have called precompute() at least once in the lifetime of this program before calling cachedComp(). For generic Perm<n>, precompute() does not affect compositions; however, for other Perm<n> classes a failure to do this will almost certainly crash your program.

Parameters

q	the first permutation to compose this with.
r	the second permutation to compose this with.

Returns: the composition of both permutations.

◆ cachedConjugate()

template<int n>

Perm< n > regina::Perm< n >::cachedConjugate ( const Perm< n > & q ) const

inline

An alias for conjugate(), provided to assist with writing generic code.

This generic Perm<n> class does not use precomputation to compute conjugates. The only point of having cachedConjugate() in this generic Perm<n> class is to make it easier to write generic code that works with Perm<n> for any n.

If you know you are only working with the generic Perm<n>, you should just call conjugate() instead.
If you are writing generic code, you must remember to call precompute() at least once in the lifetime of this program before using cachedConjugate().

Precondition: You must have called precompute() at least once in the lifetime of this program before calling cachedConjugate(). For generic Perm<n>, precompute() does not affect conjugate computations; however, for other Perm<n> classes a failure to do this will almost certainly crash your program.

Parameters

q	the permutation to conjugate this by.

Returns: the conjugate of this permutation by q.

◆ cachedInverse()

template<int n>

Perm< n > regina::Perm< n >::cachedInverse

inline

Finds the inverse of this permutation, optimised using precomputed "partial lookup tables".

The advantage of this routine is speed: calling cachedInverse() involves two table lookups and some simple arithmetic to combine the results, whereas inverse() requires time linear in n.

The disadvantages of this routine are that (1) you must remember to call precompute() in advance, and (2) the precomputed lookup tables will consume additional memory for the lifetime of your program. See precompute() for details on just how much memory these tables will consume for each n.

The permutation that is returned is the same as you would obtain by calling inverse().

Precondition: You must have called the routine precompute() at least once in the lifetime of the program before using cachedInverse(). Otherwise this routine will almost certainly crash your program.

Returns: the inverse of this permutation.

◆ cachedOrder()

template<int n>

int regina::Perm< n >::cachedOrder

inline

An alias for order(), provided to assist with writing generic code.

This specialised Perm<n> class does not use precomputation to compute orders. The only point of having cachedOrder() in this generic Perm<n> class is to make it easier to write generic code that works with Perm<n> for any n.

If you know you are only working with the generic Perm<n>, you should just call order() instead.
If you are writing generic code, you must remember to call precompute() at least once in the lifetime of this program before using cachedOrder().

Precondition: You must have called precompute() at least once in the lifetime of this program before calling cachedOrder(). For generic Perm<n>, precompute() does not affect order computations; however, for other Perm<n> classes a failure to do this will almost certainly crash your program.

Returns: the order of this permutation.

◆ cachedPow()

template<int n>

Perm< n > regina::Perm< n >::cachedPow ( long exp ) const

inline

An alias for pow(), provided to assist with writing generic code.

This specialised Perm<n> class does not use precomputation to compute powers. The only point of having cachedPow() in this generic Perm<n> class is to make it easier to write generic code that works with Perm<n> for any n.

If you know you are only working with the generic Perm<n>, you should just call pow() instead.
If you are writing generic code, you must remember to call precompute() at least once in the lifetime of this program before using cachedPow().

Precondition: You must have called precompute() at least once in the lifetime of this program before calling cachedPow(). For generic Perm<n>, precompute() does not affect powers; however, for other Perm<n> classes a failure to do this will almost certainly crash your program.

Parameters

exp	the exponent; this may be positive, zero or negative.

Returns: this permutation raised to the power of exp.

◆ clear()

template<int n>

void regina::Perm< n >::clear ( unsigned from )

inline

Resets the images of all integers from from onwards to the identity map.

Specifically, for each i in the range from,...,n-1, this routine will ensure that image[i] == i. The images of 0,1,...,from-1 will not be altered.

Precondition: The images of from,...,n-1 are exactly from,...,n-1, but possibly in a different order.

Parameters

from	the first integer whose image should be reset. This must be between 0 and n inclusive.

◆ compareWith()

template<int n>

constexpr int regina::Perm< n >::compareWith ( const Perm< n > & other ) const

constexpr

Lexicographically compares the images of (0,1,...,n-1) under this and the given permutation.

Parameters

other the permutation with which to compare this.

Returns: -1 if this permutation produces a smaller image, 0 if the permutations are equal, and 1 if this permutation produces a greater image.

◆ conjugate()

template<int n>

constexpr Perm< n > regina::Perm< n >::conjugate ( const Perm< n > & q ) const

inlineconstexpr

Computes the conjugate of this permutation by q.

Specifically, calling p.conjugate(q) is equivalent to computing q * p * q.inverse(). The resulting permutation will have the same cycle structure as p, but with the cycle elements translated according to q.

Parameters

q	the permutation to conjugate this by.

Returns: the conjugate of this permutation by q.

◆ contract()

template<int n>

template<int k>

static constexpr Perm regina::Perm< n >::contract ( Perm< k > p )

staticconstexpr

Restricts a k-element permutation to an n-element permutation, where k > n.

The resulting permutation will map 0,...,n-1 to their respective images under p, and will ignore the "unused" images p[n],...,p[k-1].

Precondition: The given permutation maps 0,...,n-1 to 0,...,n-1 in some order.

Template Parameters

k	the number of elements for the input permutation; this must be strictly greater than n.

Parameters

p	a permutation on k elements.

Returns: the same permutation restricted to a permutation on n elements.

◆ extend()

template<int n>

template<int k>

static constexpr Perm regina::Perm< n >::extend ( Perm< k > p )

staticconstexpr

Extends a k-element permutation to an n-element permutation, where 2 ≤ k < n.

The resulting permutation will map 0,...,k-1 to their respective images under p, and will map the "unused" elements k,...,n-1 to themselves.

Template Parameters

k	the number of elements for the input permutation; this must be at least 2, and strictly less than n.

Parameters

p	a permutation on k elements.

Returns: the same permutation expressed as a permutation on n elements.

◆ fromImagePack()

template<int n>

constexpr Perm< n > regina::Perm< n >::fromImagePack ( ImagePack pack )

inlinestaticconstexpr

Creates a permutation from the given image pack.

See the class notes for more information on image packs, and how they are used to build permutation codes.

For n ≥ 8, this routine is identical to fromPermCode().

Precondition: The argument pack is a valid image pack; see isImagePack() for details.

Parameters

pack	an image pack that describes a permutation.

Returns: the permutation represented by the given image pack.

◆ fromPermCode()

template<int n>

constexpr Perm< n > regina::Perm< n >::fromPermCode ( Code code )

inlinestaticconstexpr

Creates a permutation from the given internal code.

Precondition: the given code is a valid permutation code; see isPermCode() for details.

Parameters

code	the internal code for the new permutation.

Returns: the permutation reprsented by the given internal code.

◆ imagePack()

template<int n>

constexpr Perm< n >::ImagePack regina::Perm< n >::imagePack

inlineconstexpr

Returns the image pack that represents this permutation.

See the class notes for more information on image packs, and how they are used to build permutation codes.

For n ≥ 8, this routine is identical to permCode().

Returns: the image pack for this permutation.

◆ inverse()

template<int n>

constexpr Perm< n > regina::Perm< n >::inverse

inlineconstexpr

Finds the inverse of this permutation.

For permutations of seven and fewer objects, inversion is extremely fast because it uses hard-coded lookup tables. However, for this generic Perm<n> class, inversion cannot uses these lookup tables (for multiple reasons), and so inverse() takes time linear in n.

If you are going to make significant use of the generic Perm<n> class for some particular value of n, you should instead:

call precompute() to precompute some "partial lookup tables" in advance (see precompute() for details on how much memory they will consume); and then
call cachedInverse() instead of inverse() to compute your inverses.

Returns: the inverse of this permutation.

◆ isConjugacyMinimal()

template<int n>

constexpr bool regina::Perm< n >::isConjugacyMinimal

constexpr

Is this permutation minimal in its conjugacy class?

Here "minimal" means that, amongst all its conjugates, this permutation has the smallest index in the array Perm<n>::Sn.

See Sn for further information on how permutations are indexed.

Returns: true if and only if this permutation is minimal in its conjugacy class.

◆ isIdentity()

template<int n>

constexpr bool regina::Perm< n >::isIdentity

inlineconstexpr

Determines if this is the identity permutation.

This is true if and only if every integer 0 ≤ i < n is mapped to itself.

Returns: true if and only if this is the identity permutation.

◆ isImagePack()

template<int n>

constexpr bool regina::Perm< n >::isImagePack ( ImagePack pack )

inlinestaticconstexpr

Determines whether the given argument is the image pack of some n-element permutation.

See the class notes for more information on image packs, and how they are used to build permutation codes.

For n ≥ 8, this routine is identical to isPermCode().

Parameters

pack	the candidate image pack to test.

Returns: true if and only if pack is a valid image pack.

◆ isPermCode()

template<int n>

constexpr bool regina::Perm< n >::isPermCode ( Code code )

staticconstexpr

Determines whether the given integer is a valid internal permutation code.

Valid permutation codes can be passed to setPermCode() or fromPermCode(), and are returned by permCode().

Parameters

code	the permutation code to test.

Returns: true if and only if the given code is a valid internal permutation code.

◆ operator!=()

template<int n>

constexpr bool regina::Perm< n >::operator!= ( const Perm< n > & other ) const

inlineconstexpr

Determines if this differs from the given permutation.

This is true if and only if the two permutations have different images for some 0 ≤ i < n.

Parameters

other the permutation with which to compare this.

Returns: true if and only if this and the given permutation differ.

◆ operator*()

template<int n>

constexpr Perm< n > regina::Perm< n >::operator* ( const Perm< n > & q ) const

inlineconstexpr

Returns the composition of this permutation with the given permutation.

If this permutation is p, the resulting permutation will be p∘q, and will satisfy (p*q)[x] == p[q[x]].

Parameters

q	the permutation to compose this with.

Returns: the composition of both permutations.

◆ operator++() [1/2]

template<int n>

Perm & regina::Perm< n >::operator++ ( )

A preincrement operator that changes this to be the next permutation in the array Perm<n>::Sn.

If this is the last such permutation then this will wrap around to become the first permutation in Perm<n>::Sn, which is the identity.

Python: Not present. The postincrement operator is present in Python as the member function inc().

Returns: a reference to this permutation after the increment.

◆ operator++() [2/2]

template<int n>

Perm< n > regina::Perm< n >::operator++ ( int )

inline

A postincrement operator that changes this to be the next permutation in the array Perm<n>::Sn.

If this is the last such permutation then this will wrap around to become the first permutation in Perm<n>::Sn, which is the identity.

Python: This routine is named inc() since python does not support the increment operator.

Returns: a copy of this permutation before the increment took place.

◆ operator<()

template<int n>

constexpr bool regina::Perm< n >::operator< ( const Perm< n > & rhs ) const

constexpr

Determines if this appears earlier than the given permutation in the array Perm<n>::Sn.

Note that this is not the same ordering of permutations as the ordering implied by compareWith(). This is, however, consistent with the ordering implied by the ++ operators.

Unlike the smaller permutation classes that use Sn indices as internal permutation codes, for this generic Perm class the ordering defined here is slower to compute than compareWith(). It is recommended that, unless you specifically need to align your ordering with Sn indices, you either (i) use compareWith() for lexicographical ordering (which is a little faster), or else (ii) just compare permutation codes if you are happy with an arbitrary ordering (which will be much faster).

Parameters

rhs	the permutation to compare this against.

Returns: true if and only if this appears before rhs in Sn.

◆ operator=()

template<int n>

Perm & regina::Perm< n >::operator= ( const Perm< n > & cloneMe )

default

Sets this permutation to be equal to the given permutation.

Parameters

cloneMe the permutation whose value will be assigned to this permutation.

Returns: a reference to this permutation.

◆ operator==()

template<int n>

constexpr bool regina::Perm< n >::operator== ( const Perm< n > & other ) const

inlineconstexpr

Determines if this is equal to the given permutation.

This is true if and only if both permutations have the same images for all 0 ≤ i < n.

Parameters

other the permutation with which to compare this.

Returns: true if and only if this and the given permutation are equal.

◆ operator[]()

template<int n>

constexpr int regina::Perm< n >::operator[] ( int source ) const

inlineconstexpr

Determines the image of the given integer under this permutation.

Parameters

source the integer whose image we wish to find. This should be between 0 and n-1 inclusive.

Returns: the image of source.

◆ order()

template<int n>

constexpr int regina::Perm< n >::order

constexpr

Returns the order of this permutation.

In other words; this routine returns the smallest positive integer k for which the kth power of this permutation is the identity.

Note that the largest possible order for the largest supported n (n = 16) is 140. See OEIS sequence A000793 for details.

Returns: the order of this permutation.

◆ orderedSnIndex()

template<int n>

constexpr Perm< n >::Index regina::Perm< n >::orderedSnIndex

constexpr

Returns the lexicographical index of this permutation.

This will be the index of this permutation in the Perm<n>::orderedSn array.

See orderedSn for further information on lexicographical ordering.

Returns: the lexicographical index of this permutation. This will be between 0 and n!-1 inclusive.

◆ permCode()

template<int n>

constexpr Perm< n >::Code regina::Perm< n >::permCode

inlineconstexpr

Returns the internal code representing this permutation.

Note that the internal code is sufficient to reproduce the entire permutation.

The code returned will be a valid permutation code as determined by isPermCode().

Returns: the internal code.

◆ pow()

template<int n>

constexpr Perm< n > regina::Perm< n >::pow ( long exp ) const

constexpr

Computes the given power of this permutation.

This routine runs in time linear in n (in particular, the running time does not depend upon the given exponent).

Parameters

exp	the exponent; this may be positive, zero or negative.

Returns: this permutation raised to the power of exp.

◆ pre()

template<int n>

constexpr int regina::Perm< n >::pre ( int image ) const

inlineconstexpr

Determines the preimage of the given integer under this permutation.

Parameters

image the integer whose preimage we wish to find. This should be between 0 and n-1 inclusive.

Returns: the preimage of image.

◆ precompute()

template<int n>

static void regina::Perm< n >::precompute ( )

static

Performs the precomputation necessary for using the optimised cachedInverse() routine.

This must be called before calling cachedInverse().

This only needs to be done once in the lifetime of the program. If you do try to call precompute() a second time then it will do nothing and return immediately.

The precomputed tables will consume roughly:

33 kB for n = 8;
8.9 MB for n = 9;
17 MB for n = 10;
143 MB for n = 11;
268 MB for n = 12;
2.3 GB for n = 13;
4.3 GB for n = 14;
37 GB for n = 15;
69 GB for n = 16.

In particular, a 32-bit machine will not be able to store these tables for n ≥ 13, a 24-bit machine will not be these tables for n ≥ 9, and a 16-bit machine will not be able to store these tables for any n ≥ 8.

Precondition: There is enough memory available to store the precomputed tables; see above for the estimated space requirements.

Exceptions

FailedPrecondition There was not enough memory to available to store the precomputed tables.

This routine is thread-safe.

◆ rand() [1/2]

template<int n>

Perm< n > regina::Perm< n >::rand ( bool even = false )

static

Returns a random permutation on n elements.

All permutations are returned with equal probability.

This routine is thread-safe, and uses RandomEngine for its random number generation.

Warning: This routine is expensive, since it locks and unlocks the mutex protecting Regina's global uniform random bit generator. If you are calling this many times in quick succession, consider creating a single RandomEngine object yourself and then calling rand(randomEngine.engine(), even).

Parameters

even	if `true`, then the resulting permutation is guaranteed to be even (and again all even permutations are returned with equal probability).

Returns: a random permutation.

◆ rand() [2/2]

template<int n>

template<class URBG >

static Perm regina::Perm< n >::rand	(	URBG &&	gen,
		bool	even = `false`
	)

static

Returns a random permutation on n elements, using the given uniform random bit generator.

All permutations are returned with equal probability.

The thread safety of this routine is of course dependent on the thread safety of your uniform random bit generator gen.

Template Parameters

URBG	A type which, once any references are removed, must adhere to the C++ UniformRandomBitGenerator concept.

Python: Not present. Python users are still able to use the non-thread-safe variant without the gen argument.

Parameters

gen	the source of randomness to use (e.g., one of the many options provided in the C++ standard `random` header).
even	if `true`, then the resulting permutation is guaranteed to be even (and again all even permutations are returned with equal probability).

Returns: a random permutation.

◆ reverse()

template<int n>

constexpr Perm< n > regina::Perm< n >::reverse

inlineconstexpr

Finds the reverse of this permutation.

Here reverse means that we reverse the images of 0,...,n-1. In other words, if permutation q is the reverse of p, then p[i] == q[n - 1 - i] for all i.

◆ rot()

template<int n>

constexpr Perm< n > regina::Perm< n >::rot ( int i )

staticconstexpr

Returns the ith rotation.

This maps k to k + i (mod n) for all k.

Parameters

i	the image of 0; this must be between 0 and n-1 inclusive.

Returns: the ith rotation.

◆ setPermCode()

template<int n>

void regina::Perm< n >::setPermCode ( Code code )

inline

Sets this permutation to that represented by the given internal code.

Precondition: the given code is a valid permutation code; see isPermCode() for details.

Parameters

code	the internal code that will determine the new value of this permutation.

◆ sign()

template<int n>

constexpr int regina::Perm< n >::sign

constexpr

Determines the sign of this permutation.

Returns: 1 if this permutation is even, or -1 if this permutation is odd.

◆ SnIndex()

template<int n>

constexpr Perm< n >::Index regina::Perm< n >::SnIndex

constexpr

Returns the index of this permutation in the Perm<n>::Sn array.

See Sn for further information on how these permutations are indexed.

Returns: the index i for which this permutation is equal to Perm<n>::Sn[i]. This will be between 0 and n!-1 inclusive.

◆ str()

template<int n>

std::string regina::Perm< n >::str

Returns a string representation of this permutation.

The representation will consist of n adjacent digits representing the images of 0,...,n-1 respectively. If n > 10, then lower-case hexadecimal digits will be used.

An example of a string representation for n = 5 is 30421.

Returns: a string representation of this permutation.

◆ tightDecode()

template<int n>

Perm< n > regina::Perm< n >::tightDecode ( std::istream & input )

inlinestatic

Reconstructs a permutation from its given tight encoding.

See the page on tight encodings for details.

The tight encoding will be read from the given input stream. If the input stream contains leading whitespace then it will be treated as an invalid encoding (i.e., this routine will throw an exception). The input routine may contain further data: if this routine is successful then the input stream will be left positioned immediately after the encoding, without skipping any trailing whitespace.

Tight encodings are fast to work with for small permutation classes (n ≤ 7), but slower for larger permutation classes (8 ≤ n ≤ 16). See tightEncoding() for further details.

Exceptions

InvalidInput The given input stream does not begin with a tight encoding of an n-element permutation.

Python: Not present. Use tightDecoding() instead, which takes a string as its argument.

Parameters

input an input stream that begins with the tight encoding for an n-element permutation.

Returns: the permutation represented by the given tight encoding.

◆ tightDecoding()

template<int n>

Perm< n > regina::Perm< n >::tightDecoding ( const std::string & enc )

inlinestatic

Reconstructs a permutation from its given tight encoding.

See the page on tight encodings for details.

The tight encoding will be given as a string. If this string contains leading whitespace or any trailing characters at all (including trailing whitespace), then it will be treated as an invalid encoding (i.e., this routine will throw an exception).

Tight encodings are fast to work with for small permutation classes (n ≤ 7), but slower for larger permutation classes (8 ≤ n ≤ 16). See tightEncoding() for further details.

Exceptions

InvalidArgument The given string is not a tight encoding of an n-element permutation.

Parameters

enc	the tight encoding for an n-element permutation.

Returns: the permutation represented by the given tight encoding.

◆ tightEncode()

template<int n>

void regina::Perm< n >::tightEncode ( std::ostream & out ) const

Writes the tight encoding of this permutation to the given output stream.

See the page on tight encodings for details.

For all permutation classes Perm<n>, the tight encoding is based on the index into the full permutation group S_n. For smaller permutation classes (n ≤ 7), such encodings are very fast to work with since the S_n index is used as the internal permutation code. For larger permutation classes however (8 ≤ n ≤ 16), the S_n index requires some non-trivial work to compute.

Python: Not present. Use tightEncoding() instead, which returns a string.

Parameters

out	the output stream to which the encoded string will be written.

◆ tightEncoding()

template<int n>

std::string regina::Perm< n >::tightEncoding

Returns the tight encoding of this permutation.

See the page on tight encodings for details.

For all permutation classes Perm<n>, the tight encoding is based on the index into the full permutation group S_n. For smaller permutation classes (n ≤ 7), such encodings are very fast to work with since the S_n index is used as the internal permutation code. For larger permutation classes however (8 ≤ n ≤ 16), the S_n index requires some non-trivial work to compute.

Returns: the resulting encoded string.

◆ trunc()

template<int n>

std::string regina::Perm< n >::trunc ( int len ) const

Returns a prefix of the string representation of this permutation, containing only the images of the first len integers.

Parameters

len	the length of the prefix required; this must be between 0 and n inclusive.

Returns: the corresponding prefix of the string representation of this permutation.

Member Data Documentation

◆ code_

template<int n>

Code regina::Perm< n >::code_

protected

The internal code representing this permutation.

◆ codeType

template<int n>

constexpr PermCodeType regina::Perm< n >::codeType = PERM_CODE_IMAGES

staticconstexpr

Indicates what type of internal permutation code is used by this instance of the Perm class template.

◆ imageBits

template<int n>

constexpr int regina::Perm< n >::imageBits = regina::bitsRequired(n)

staticconstexpr

Indicates the number of bits used in an image pack to store the image of a single integer.

A full image pack combines n such images together, and so uses n * imageBits bits in total.

◆ imageMask

template<int n>

constexpr ImagePack regina::Perm< n >::imageMask

staticconstexpr

Initial value:

=

(static_cast<ImagePack>(1) << Perm<n>::imageBits) - 1

regina::Perm::ImagePack

typename IntOfMinSize<(imageBits *n+7)/8 >::utype ImagePack

Indicates the native unsigned integer type used to store a single image pack.

Definition: perm.h:279

regina::Perm::imageBits

static constexpr int imageBits

Indicates the number of bits used in an image pack to store the image of a single integer.

Definition: perm.h:257

A bitmask whose lowest imageBits bits are 1, and whose remaining higher order bits are all 0.

This may be useful when creating or analysing image packs.

◆ nPerms

template<int n>

constexpr Index regina::Perm< n >::nPerms = factorial(n)

staticconstexpr

The total number of permutations on n elements.

This is the size of the symmetric group S_n.

◆ nPerms_1

template<int n>

constexpr Index regina::Perm< n >::nPerms_1 = factorial(n-1)

staticconstexpr

The total number of permutations on n-1 elements.

This is the size of the symmetric group S_n-1.

◆ orderedSn

template<int n>

constexpr OrderedSnLookup regina::Perm< n >::orderedSn {}

staticconstexpr

Gives array-like access to all possible permutations of n elements in lexicographical order.

To access the permutation at index i, you simply use the square bracket operator: orderedSn[i]. The index i must be between 0 and n!-1 inclusive.

Lexicographical ordering treats each permutation p as the n-tuple (p[0], p[1], ..., p[n-1]).

This array is different from Perm<n>::Sn, since orderedSn stores permutations in lexicographical order, whereas Sn alternates between even and odd permutations.

This is a lightweight object, and it is defined in the headers only. In particular, you cannot make a reference to it (but it is cheap to make a copy).

Warning: For n ≤ 7, the square bracket operator is a very fast constant-time routine. However, for n ≥ 8, this is not constant time; the current implementation is quadratic in n.

◆ Sn

template<int n>

constexpr SnLookup regina::Perm< n >::Sn {}

staticconstexpr

Gives array-like access to all possible permutations of n elements.

To access the permutation at index i, you simply use the square bracket operator: Sn[i]. The index i must be between 0 and n!-1 inclusive.

The permutations with even indices in the array are the even permutations, and those with odd indices in the array are the odd permutations.

This array is different from Perm<n>::orderedSn, since Sn alternates between even and odd permutations, whereas orderedSn stores permutations in lexicographical order.

This is a lightweight object, and it is defined in the headers only. In particular, you cannot make a reference to it (but it is cheap to make a copy).

Warning: For n ≤ 7, the square bracket operator is a very fast constant-time routine. However, for n ≥ 8, this is not constant time; the current implementation is quadratic in n.

The documentation for this class was generated from the following file:

maths/perm.h

Public Types

Public Member Functions

Static Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ Code

◆ ImagePack

◆ Index

Constructor & Destructor Documentation

◆ Perm() [1/5]

◆ Perm() [2/5]

◆ Perm() [3/5]

◆ Perm() [4/5]

◆ Perm() [5/5]

Member Function Documentation

◆ cachedComp() [1/2]

◆ cachedComp() [2/2]

◆ cachedConjugate()

◆ cachedInverse()

◆ cachedOrder()

◆ cachedPow()

◆ clear()

◆ compareWith()

◆ conjugate()

◆ contract()

◆ extend()

◆ fromImagePack()

◆ fromPermCode()

◆ imagePack()

◆ inverse()

◆ isConjugacyMinimal()

◆ isIdentity()

◆ isImagePack()

◆ isPermCode()

◆ operator!=()

◆ operator*()

◆ operator++() [1/2]

◆ operator++() [2/2]

◆ operator<()

◆ operator=()

◆ operator==()

◆ operator[]()

◆ order()

◆ orderedSnIndex()

◆ permCode()

◆ pow()

◆ pre()

◆ precompute()

◆ rand() [1/2]

◆ rand() [2/2]

◆ reverse()

◆ rot()

◆ setPermCode()

◆ sign()

◆ SnIndex()

◆ str()

◆ tightDecode()

◆ tightDecoding()

◆ tightEncode()

◆ tightEncoding()

◆ trunc()

Member Data Documentation

◆ code_

◆ codeType

◆ imageBits

◆ imageMask

◆ nPerms

◆ nPerms_1

◆ orderedSn

◆ Sn