Regina Calculation Engine
Public Types | Public Member Functions | Static Public Member Functions | Static Public Attributes | List of all members
regina::NPerm< 3 > Class Template Reference

Represents a permutation of {0,1,2}. More...

#include <maths/nperm3.h>

Public Types

enum  {
  code012 = 0, code021 = 1, code120 = 2, code102 = 3,
  code201 = 4, code210 = 5
}
 
typedef int Index
 Denotes a native signed integer type large enough to count all permutations on three elements. More...
 
typedef uint8_t Code
 Indicates the native unsigned integer type used to store the internal permutation code. More...
 

Public Member Functions

 NPerm ()
 Creates the identity permutation. More...
 
 NPerm (int a, int b)
 Creates the transposition of a and b. More...
 
 NPerm (int a, int b, int c)
 Creates a permutation mapping (0,1,2) to (a,b,c) respectively. More...
 
 NPerm (const int *image)
 Creates a permutation mapping i to image[i] for each i = 0,1,2. More...
 
 NPerm (const int *a, const int *b)
 Creates a permutation mapping (a[0], ..., a[2]) to (b[0], ..., b[2]) respectively. More...
 
 NPerm (const NPerm< 3 > &cloneMe)
 Creates a permutation that is a clone of the given permutation. More...
 
Code permCode () const
 Returns the internal code representing this permutation. More...
 
REGINA_DEPRECATED Code getPermCode () const
 Deprecated routine that returns the internal code representing this permutation. More...
 
void setPermCode (Code code)
 Sets this permutation to that represented by the given internal code. More...
 
NPerm< 3 > & operator= (const NPerm< 3 > &cloneMe)
 Sets this permutation to be equal to the given permutation. More...
 
NPerm< 3 > operator* (const NPerm< 3 > &q) const
 Returns the composition of this permutation with the given permutation. More...
 
NPerm< 3 > inverse () const
 Finds the inverse of this permutation. More...
 
NPerm< 3 > reverse () const
 Finds the reverse of this permutation. More...
 
int sign () const
 Determines the sign of this permutation. More...
 
int operator[] (int source) const
 Determines the image of the given integer under this permutation. More...
 
int preImageOf (int image) const
 Determines the preimage of the given integer under this permutation. More...
 
bool operator== (const NPerm< 3 > &other) const
 Determines if this is equal to the given permutation. More...
 
bool operator!= (const NPerm< 3 > &other) const
 Determines if this differs from the given permutation. More...
 
int compareWith (const NPerm< 3 > &other) const
 Lexicographically compares the images of (0,1,2) under this and the given permutation. More...
 
bool isIdentity () const
 Determines if this is the identity permutation. More...
 
Index index () const
 Returns the lexicographical index of this permutation. More...
 
std::string str () const
 Returns a string representation of this permutation. More...
 
std::string trunc (unsigned len) const
 Returns a prefix of the string representation of this permutation, containing only the images of the first len integers. More...
 
std::string trunc2 () const
 Returns a string representation of this permutation with only the images of 0 and 1. More...
 
int S3Index () const
 Returns the index of this permutation in the NPerm<3>::S3 array. More...
 
int SnIndex () const
 Returns the index of this permutation in the NPerm<3>::S3 array. More...
 
REGINA_INLINE_REQUIRED int orderedS3Index () const
 Returns the index of this permutation in the NPerm<3>::orderedS3 array. More...
 
int orderedSnIndex () const
 Returns the index of this permutation in the NPerm<3>::orderedS3 array. More...
 

Static Public Member Functions

static NPerm< 3 > fromPermCode (Code code)
 Creates a permutation from the given internal code. More...
 
static bool isPermCode (Code code)
 Determines whether the given integer is a valid internal permutation code. More...
 
static NPerm atIndex (Index i)
 Returns the ith permutation on three elements, where permutations are numbered lexicographically beginning at 0. More...
 
static NPerm rand ()
 Returns a random permutation on three elements. More...
 
template<int k>
static NPerm< 3 > extend (NPerm< k > p)
 Extends a k-element permutation to an 3-element permutation. More...
 
template<int k>
static NPerm< 3 > contract (NPerm< k > p)
 Restricts a k-element permutation to an 3-element permutation, where k > 3. More...
 

Static Public Attributes

static const Index nPerms = 6
 The total number of permutations on three elements. More...
 
static const Index nPerms_1 = 2
 The total number of permutations on two elements. More...
 
static const NPerm< 3 > S3 [6]
 Contains all possible permutations of three elements. More...
 
static const NPerm< 3 > * Sn
 A dimension-agnostic alias for NPerm<3>::S3. More...
 
static const unsigned invS3 [6]
 Contains the inverses of the permutations in the array S3. More...
 
static const unsigned * invSn
 A dimension-agnostic alias for NPerm<3>::invS3. More...
 
static const NPerm< 3 > orderedS3 [6]
 Contains all possible permutations of three elements in lexicographical order. More...
 
static const NPerm< 3 > * orderedSn
 A dimension-agnostic alias for NPerm<3>::orderedS3. More...
 
static const NPerm< 3 > S2 [2]
 Contains all possible permutations of two elements. More...
 
static const NPerm< 3 > * Sn_1
 A dimension-agnostic alias for NPerm<3>::S2. More...
 

Detailed Description

template<>
class regina::NPerm< 3 >

Represents a permutation of {0,1,2}.

This is a specialisation of the generic NPerm template: it is highly optimised, and also offers some additional functionality. Amongst other things, this permutation class is used to specify how simplices of a 2-manifold triangulation are glued together.

As with all NPerm template classes, these objects are small enough to pass about by value instead of by reference. Moreover, NPerm3 in particular is extremely fast to work with.

Each permutation has an internal code, which is a single native integer that is sufficient to reconstruct the permutation. Thus the internal code may be a useful means for passing permutation objects to and from the engine. For NPerm3, the internal code is an integer between 0 and 5 inclusive that gives the index of the permutation in the array NPerm<3>::S3.

Python:
Since Python does not support templates, this class is made available under the name NPerm3.

Member Typedef Documentation

§ Code

typedef uint8_t regina::NPerm< 3 >::Code

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

§ Index

typedef int regina::NPerm< 3 >::Index

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

In other words, this is a native signed integer type large enough to store (3!).

Member Enumeration Documentation

§ anonymous enum

anonymous enum
Enumerator
code012 

The internal code for the permutation (0,1,2).

Python:
Not present.
code021 

The internal code for the permutation (0,2,1).

Python:
Not present.
code120 

The internal code for the permutation (1,2,0).

Python:
Not present.
code102 

The internal code for the permutation (1,0,2).

Python:
Not present.
code201 

The internal code for the permutation (2,0,1).

Python:
Not present.
code210 

The internal code for the permutation (2,1,0).

Python:
Not present.

Constructor & Destructor Documentation

§ NPerm() [1/6]

regina::NPerm< 3 >::NPerm ( )
inline

Creates the identity permutation.

§ NPerm() [2/6]

regina::NPerm< 3 >::NPerm ( int  a,
int  b 
)
inline

Creates the transposition of a and b.

Note that a and b need not be distinct.

Precondition
a and b are in {0,1,2}.
Parameters
athe element to switch with b.
bthe element to switch with a.

§ NPerm() [3/6]

regina::NPerm< 3 >::NPerm ( int  a,
int  b,
int  c 
)
inline

Creates a permutation mapping (0,1,2) to (a,b,c) respectively.

Precondition
{a,b,c} = {0,1,2}.
Parameters
athe desired image of 0.
bthe desired image of 1.
cthe desired image of 2.

§ NPerm() [4/6]

regina::NPerm< 3 >::NPerm ( const int *  image)
inline

Creates a permutation mapping i to image[i] for each i = 0,1,2.

Precondition
The array image contains three elements, which are 0, 1 and 2 in some order.
Python:
Not present.
Parameters
imagethe array of images.

§ NPerm() [5/6]

regina::NPerm< 3 >::NPerm ( const int *  a,
const int *  b 
)

Creates a permutation mapping (a[0], ..., a[2]) to (b[0], ..., b[2]) respectively.

Precondition
Both arrays a and b contain 3 elements, which are 0,...,2 in some order.
Python:
Not present.
Parameters
athe array of preimages; this must have length 3.
bthe corresponding array of images; this must also have length 3.

§ NPerm() [6/6]

regina::NPerm< 3 >::NPerm ( const NPerm< 3 > &  cloneMe)
inline

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

Parameters
cloneMethe permutation to clone.

Member Function Documentation

§ atIndex()

NPerm< 3 > regina::NPerm< 3 >::atIndex ( Index  i)
inlinestatic

Returns the ith permutation on three elements, where permutations are numbered lexicographically beginning at 0.

Lexicographical ordering treats each permutation p as the 3-tuple (p[0], p[1], p[2]).

The return value will be identical to orderedS3[i].

Parameters
ithe lexicographical index of the permutation; this must be between 0 and 5 inclusive.
Returns
the ith permutation.

§ compareWith()

int regina::NPerm< 3 >::compareWith ( const NPerm< 3 > &  other) const
inline

Lexicographically compares the images of (0,1,2) under this and the given permutation.

Parameters
otherthe 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.

§ contract()

template<int k>
static NPerm<3> regina::NPerm< 3 >::contract ( NPerm< k >  p)
static

Restricts a k-element permutation to an 3-element permutation, where k > 3.

The resulting permutation will map 0,1,2 to their respective images under p, and will ignore the "unused" images p[3],...,p[k-1].

Precondition
The given permutation maps 0,1,2 to 0,1,2 in some order.
Template Parameters
kthe number of elements for the input permutation; this must be strictly greater than 3.
Parameters
pa permutation on k elements.
Returns
the same permutation restricted to a permutation on 3 elements.

§ extend()

template<int k>
static NPerm<3> regina::NPerm< 3 >::extend ( NPerm< k >  p)
static

Extends a k-element permutation to an 3-element permutation.

where 2 ≤ k < 3. The only possible value of k is 2, but this routine is kept as a template function for consistency with the other classes' NPerm<n>::extend() routines.

The resulting permutation will map 0,1 to their respective images under p, and will map the "unused" element 3 to itself.

Template Parameters
kthe number of elements for the input permutation; this must be exactly 2.
Parameters
pa permutation on two elements.
Returns
the same permutation expressed as a permutation on three elements.

§ fromPermCode()

NPerm< 3 > regina::NPerm< 3 >::fromPermCode ( Code  code)
inlinestatic

Creates a permutation from the given internal code.

Precondition
the given code is a valid permutation code; see isPermCode() for details.
Parameters
codethe internal code for the new permutation.
Returns
the permutation represented by the given internal code.

§ getPermCode()

NPerm< 3 >::Code regina::NPerm< 3 >::getPermCode ( ) const
inline

Deprecated routine that returns the internal code representing this permutation.

Deprecated:
This routine has been renamed to permCode(). See the permCode() documentation for further details.

§ index()

NPerm< 3 >::Index regina::NPerm< 3 >::index ( ) const
inline

Returns the lexicographical index of this permutation.

This indicates where this permutation sits within a full lexicographical ordering of all 3! permutations on three elements.

Lexicographical ordering treats each permutation p as the 3-tuple (p[0], p[1], p[2]). In particular, the identity permutation has index 0, and the "reverse" permutation (which maps each i to 2-i) has index 5 = 3!-1.

This routine is identical to orderedS3Index().

Returns
the index of this permutation, which will be between 0 and 5 inclusive.

§ inverse()

NPerm< 3 > regina::NPerm< 3 >::inverse ( ) const
inline

Finds the inverse of this permutation.

Returns
the inverse of this permutation.

§ isIdentity()

bool regina::NPerm< 3 >::isIdentity ( ) const
inline

Determines if this is the identity permutation.

This is true if and only if each of 0, 1 and 2 is mapped to itself.

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

§ isPermCode()

bool regina::NPerm< 3 >::isPermCode ( Code  code)
inlinestatic

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().

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

§ operator!=()

bool regina::NPerm< 3 >::operator!= ( const NPerm< 3 > &  other) const
inline

Determines if this differs from the given permutation.

This is true if and only if the two permutations have different images for at least one of 0, 1 or 2.

Parameters
otherthe permutation with which to compare this.
Returns
true if and only if this and the given permutation differ.

§ operator*()

NPerm< 3 > regina::NPerm< 3 >::operator* ( const NPerm< 3 > &  q) const
inline

Returns the composition of this permutation with the given permutation.

If this permutation is p, the resulting permutation will be p o q, satisfying (p*q)[x] == p[q[x]].

Parameters
qthe permutation with which to compose this.
Returns
the composition of both permutations.

§ operator=()

NPerm< 3 > & regina::NPerm< 3 >::operator= ( const NPerm< 3 > &  cloneMe)
inline

Sets this permutation to be equal to the given permutation.

Parameters
cloneMethe permutation whose value will be assigned to this permutation.
Returns
a reference to this permutation.

§ operator==()

bool regina::NPerm< 3 >::operator== ( const NPerm< 3 > &  other) const
inline

Determines if this is equal to the given permutation.

This is true if and only if both permutations have the same images for 0, 1 and 2.

Parameters
otherthe permutation with which to compare this.
Returns
true if and only if this and the given permutation are equal.

§ operator[]()

int regina::NPerm< 3 >::operator[] ( int  source) const
inline

Determines the image of the given integer under this permutation.

Parameters
sourcethe integer whose image we wish to find. This should be between 0 and 2 inclusive.
Returns
the image of source.

§ orderedS3Index()

int regina::NPerm< 3 >::orderedS3Index ( ) const
inline

Returns the index of this permutation in the NPerm<3>::orderedS3 array.

Returns
the index i for which this permutation is equal to NPerm<3>::orderedS3[i]. This will be between 0 and 5 inclusive.

§ orderedSnIndex()

int regina::NPerm< 3 >::orderedSnIndex ( ) const
inline

Returns the index of this permutation in the NPerm<3>::orderedS3 array.

This is a dimension-agnostic alias for orderedS3Index().

Returns
the index i for which this permutation is equal to NPerm<3>::orderedS3[i]. This will be between 0 and 5 inclusive.

§ permCode()

NPerm< 3 >::Code regina::NPerm< 3 >::permCode ( ) const
inline

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.

§ preImageOf()

int regina::NPerm< 3 >::preImageOf ( int  image) const
inline

Determines the preimage of the given integer under this permutation.

Parameters
imagethe integer whose preimage we wish to find. This should be between 0 and 2 inclusive.
Returns
the preimage of image.

§ rand()

NPerm< 3 > regina::NPerm< 3 >::rand ( )
inlinestatic

Returns a random permutation on three elements.

All permutations are returned with equal probability.

The implementation uses the C standard ::rand() function for its random number generation.

Returns
a random permutation.

§ reverse()

NPerm< 3 > regina::NPerm< 3 >::reverse ( ) const
inline

Finds the reverse of this permutation.

Here reverse means that we reverse the images of 0,1,2. In other words, if permutation q is the reverse of p, then p[i] == q[2 - i] for all i.

§ S3Index()

int regina::NPerm< 3 >::S3Index ( ) const
inline

Returns the index of this permutation in the NPerm<3>::S3 array.

Returns
the index i for which this permutation is equal to NPerm<3>::S3[i]. This will be between 0 and 5 inclusive.

§ setPermCode()

void regina::NPerm< 3 >::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
codethe internal code that will determine the new value of this permutation.

§ sign()

int regina::NPerm< 3 >::sign ( ) const
inline

Determines the sign of this permutation.

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

§ SnIndex()

int regina::NPerm< 3 >::SnIndex ( ) const
inline

Returns the index of this permutation in the NPerm<3>::S3 array.

This is a dimension-agnostic alias for S3Index().

Returns
the index i for which this permutation is equal to NPerm<3>::S3[i]. This will be between 0 and 5 inclusive.

§ str()

std::string regina::NPerm< 3 >::str ( ) const

Returns a string representation of this permutation.

The representation will consist of three adjacent digits representing the images of 0, 1 and 2 respectively. An example of a string representation is 120.

Returns
a string representation of this permutation.

§ trunc()

std::string regina::NPerm< 3 >::trunc ( unsigned  len) const

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

Parameters
lenthe length of the prefix required; this must be between 0 and 3 inclusive.
Returns
the corresponding prefix of the string representation of this permutation.

§ trunc2()

std::string regina::NPerm< 3 >::trunc2 ( ) const

Returns a string representation of this permutation with only the images of 0 and 1.

The resulting string will therefore have length two.

Returns
a truncated string representation of this permutation.

Member Data Documentation

§ invS3

const unsigned regina::NPerm< 3 >::invS3[6]
static

Contains the inverses of the permutations in the array S3.

Specifically, the inverse of permutation S3[i] is the permutation S3[ invS3[i] ].

§ invSn

const unsigned* regina::NPerm< 3 >::invSn
static

A dimension-agnostic alias for NPerm<3>::invS3.

In general, for each K the class NPermK will define an alias invSn that references the list of all permutations NPermK::invSK.

§ nPerms

const Index regina::NPerm< 3 >::nPerms = 6
static

The total number of permutations on three elements.

This is the size of the array Sn.

§ nPerms_1

const Index regina::NPerm< 3 >::nPerms_1 = 2
static

The total number of permutations on two elements.

This is the size of the array Sn_1.

§ orderedS3

const NPerm<3> regina::NPerm< 3 >::orderedS3[6]
static

Contains all possible permutations of three elements in lexicographical order.

§ orderedSn

const NPerm<3>* regina::NPerm< 3 >::orderedSn
static

A dimension-agnostic alias for NPerm<3>::orderedS3.

In general, for each K the class NPermK will define an alias orderedSn that references the list of all permutations NPermK::orderedSK.

§ S2

const NPerm<3> regina::NPerm< 3 >::S2[2]
static

Contains all possible permutations of two elements.

In each permutation, 2 maps to 2.

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

For all permutation classes (NPerm<3>, NPerm4 and so on), the S2 array stores the same permutations in the same order (but of course using different data types).

Note that these permutations are already in lexicographical order.

§ S3

const NPerm<3> regina::NPerm< 3 >::S3[6]
static

Contains all possible permutations of three elements.

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

For all permutation classes (NPerm<3>, NPerm4 and so on), the S3 array stores the same permutations in the same order (but of course using different data types).

Note that these permutations are not necessarily in lexicographical order.

§ Sn

const NPerm<3>* regina::NPerm< 3 >::Sn
static

A dimension-agnostic alias for NPerm<3>::S3.

In general, for each K the class NPermK will define an alias Sn that references the list of all permutations NPermK::SK.

§ Sn_1

const NPerm<3>* regina::NPerm< 3 >::Sn_1
static

A dimension-agnostic alias for NPerm<3>::S2.

In general, for each K the class NPermK will define an alias Sn_1 that references the list of all permutations NPermK::S(K-1).


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

Copyright © 1999-2016, The Regina development team
This software is released under the GNU General Public License, with some additional permissions; see the source code for details.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@maths.uq.edu.au).