Regina Calculation Engine
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regina::NPerm4 Class Reference

Represents a permutation of {0,1,2,3}. More...

#include <maths/nperm4.h>

Public Types

enum  { nPerms = 24, nPerms_1 = 6 }
 

Public Member Functions

 NPerm4 ()
 Creates the identity permutation. More...
 
 NPerm4 (int a, int b)
 Creates the transposition of a and b. More...
 
 NPerm4 (int a, int b, int c, int d)
 Creates a permutation mapping (0,1,2,3) to (a,b,c,d) respectively. More...
 
 NPerm4 (const int *image)
 Creates a permutation mapping i to image[i] for each i = 0,1,2,3. More...
 
 NPerm4 (int a0, int a1, int b0, int b1, int c0, int c1, int d0, int d1)
 Creates a permutation mapping (a0,b0,c0,d0) to (a1,b1,c1,d1) respectively. More...
 
 NPerm4 (const NPerm4 &cloneMe)
 Creates a permutation that is a clone of the given permutation. More...
 
unsigned char getPermCode () const
 Returns the first-generation code representing this permutation. More...
 
unsigned char getPermCode2 () const
 Returns the second-generation code representing this permutation. More...
 
void setPermCode (unsigned char code)
 Sets this permutation to that represented by the given first-generation permutation code. More...
 
void setPermCode2 (unsigned char code)
 Sets this permutation to that represented by the given second-generation permutation code. More...
 
void setPerm (int a, int b)
 Sets this permutation to the transposition of a and b. More...
 
void setPerm (int a, int b, int c, int d)
 Sets this permutation to that mapping (0,1,2,3) to (a,b,c,d) respectively. More...
 
NPerm4operator= (const NPerm4 &cloneMe)
 Sets this permutation to be equal to the given permutation. More...
 
NPerm4 operator* (const NPerm4 &q) const
 Returns the composition of this permutation with the given permutation. More...
 
NPerm4 inverse () const
 Finds the inverse 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 NPerm4 &other) const
 Determines if this is equal to the given permutation. More...
 
bool operator!= (const NPerm4 &other) const
 Determines if this differs from the given permutation. More...
 
int compareWith (const NPerm4 &other) const
 Lexicographically compares the images of (0,1,2,3) under this and the given permutation. More...
 
bool isIdentity () const
 Determines if this is the identity permutation. More...
 
std::string toString () const
 A deprecated alias for str(), which returns a string representation of this permutation. More...
 
std::string str () const
 Returns a string representation of this permutation. More...
 
std::string trunc2 () const
 Returns a string representation of this permutation with only the images of 0 and 1. More...
 
std::string trunc3 () const
 Returns a string representation of this permutation with only the images of 0, 1 and 2 included. More...
 
int S4Index () const
 Returns the index of this permutation in the NPerm4::S4 array. More...
 
int SnIndex () const
 Returns the index of this permutation in the NPerm4::S4 array. More...
 
int orderedS4Index () const
 Returns the index of this permutation in the NPerm4::orderedS4 array. More...
 
int orderedSnIndex () const
 Returns the index of this permutation in the NPerm4::orderedS4 array. More...
 

Static Public Member Functions

static NPerm4 fromPermCode (unsigned char code)
 Creates a permutation from the given first-generation permutation code. More...
 
static NPerm4 fromPermCode2 (unsigned char code)
 Creates a permutation from the given second-generation permutation code. More...
 
static bool isPermCode (unsigned char code)
 Determines whether the given character is a valid first-generation permutation code. More...
 
static bool isPermCode2 (unsigned char code)
 Determines whether the given character is a valid second-generation permutation code. More...
 

Static Public Attributes

static const NPerm4 S4 [24]
 Contains all possible permutations of four elements. More...
 
static const NPerm4Sn
 A dimension-agnostic alias for NPerm4::S4. More...
 
static const unsigned invS4 [24]
 Contains the inverses of the permutations in the array S4. More...
 
static const unsigned * invSn
 A dimension-agnostic alias for NPerm4::invS4. More...
 
static const NPerm4 orderedS4 [24]
 Contains all possible permutations of four elements in lexicographical order. More...
 
static const NPerm4orderedSn
 A dimension-agnostic alias for NPerm4::orderedS4. More...
 
static const NPerm4 S3 [6]
 Contains all possible permutations of three elements. More...
 
static const NPerm4Sn_1
 A dimension-agnostic alias for NPerm4::S3. More...
 
static const unsigned invS3 [6]
 Contains the inverses of the permutations in the array S3. More...
 
static const NPerm4 orderedS3 [6]
 Contains all possible permutations of three elements in lexicographical order. More...
 
static const NPerm4 S2 [2]
 Contains all possible permutations of two elements. More...
 
static const unsigned invS2 [2]
 Contains the inverses of the permutations in the array S2. More...
 

Friends

std::ostream & operator<< (std::ostream &out, const NPerm4 &p)
 Writes a string representation of the given permutation to the given output stream. More...
 

Detailed Description

Represents a permutation of {0,1,2,3}.

Amongst other things, such permutations are used in specifying how simplices of a 3-manifold triangulation are glued together. NPerm4 objects are small enough to pass about by value instead of by reference.

Each permutation has an internal code, and this code is sufficient to reconstruct the permutation. Thus the internal code may be a useful means for passing permutation objects to and from the engine.

The internal permutation codes have changed as of Regina 4.6.1:

It is highly recommended that, if you need to work with permutation codes at all, you use second-generation codes where possible. This is because the first-generation routines incur additional overhead in converting back and forth between the second-generation codes (which are used internally by NPerm4).

Member Enumeration Documentation

anonymous enum
Enumerator
nPerms 

The total number of permutations on four elements.

This is the size of the array Sn.

Python:
Not present.
nPerms_1 

The total number of permutations on three elements.

This is the size of the array Sn_1.

Python:
Not present.

Constructor & Destructor Documentation

regina::NPerm4::NPerm4 ( )
inline

Creates the identity permutation.

regina::NPerm4::NPerm4 ( 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 {0,1,2,3}.
Parameters
athe element to switch with b.
bthe element to switch with a.
regina::NPerm4::NPerm4 ( int  a,
int  b,
int  c,
int  d 
)
inline

Creates a permutation mapping (0,1,2,3) to (a,b,c,d) respectively.

Precondition
{a,b,c,d} = {0,1,2,3}.
Parameters
athe desired image of 0.
bthe desired image of 1.
cthe desired image of 2.
dthe desired image of 3.
regina::NPerm4::NPerm4 ( const int *  image)
inline

Creates a permutation mapping i to image[i] for each i = 0,1,2,3.

Precondition
The array image contains four elements, which are 0, 1, 2 and 3 in some order.
Python:
Not present.
Parameters
imagethe array of images.
regina::NPerm4::NPerm4 ( int  a0,
int  a1,
int  b0,
int  b1,
int  c0,
int  c1,
int  d0,
int  d1 
)

Creates a permutation mapping (a0,b0,c0,d0) to (a1,b1,c1,d1) respectively.

Precondition
{a0,b0,c0,d0} = {a1,b1,c1,d1} = {0,1,2,3}.
Parameters
a0the desired preimage of a1.
b0the desired preimage of b1.
c0the desired preimage of c1.
d0the desired preimage of d1.
a1the desired image of a0.
b1the desired image of b0.
c1the desired image of c0.
d1the desired image of d0.
regina::NPerm4::NPerm4 ( const NPerm4 cloneMe)
inline

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

Parameters
cloneMethe permutation to clone.

Member Function Documentation

int regina::NPerm4::compareWith ( const NPerm4 other) const
inline

Lexicographically compares the images of (0,1,2,3) 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.
NPerm4 regina::NPerm4::fromPermCode ( unsigned char  code)
inlinestatic

Creates a permutation from the given first-generation permutation code.

Precondition
the given code is a valid first-generation permutation code; see isPermCode() for details.
Warning
This routine will incur additional overhead, since NPerm4 now uses second-generation codes internally. See the class notes and the routine fromPermCode2() for details.
Parameters
codethe first-generation code for the new permutation.
Returns
the permutation represented by the given code.
NPerm4 regina::NPerm4::fromPermCode2 ( unsigned char  code)
inlinestatic

Creates a permutation from the given second-generation permutation code.

Second-generation codes are fast to work with, since they are used internally by the NPerm4 class.

Precondition
the given code is a valid second-generation permutation code; see isPermCode2() for details.
Parameters
codethe second-generation code for the new permutation.
Returns
the permutation represented by the given code.
unsigned char regina::NPerm4::getPermCode ( ) const
inline

Returns the first-generation code representing this permutation.

This code is sufficient to reproduce the entire permutation.

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

Warning
This routine will incur additional overhead, since NPerm4 now uses second-generation codes internally. See the class notes and the routine getPermCode2() for details.
Returns
the first-generation permutation code.
unsigned char regina::NPerm4::getPermCode2 ( ) const
inline

Returns the second-generation code representing this permutation.

This code is sufficient to reproduce the entire permutation.

The code returned will be a valid second-generation permutation code as determined by isPermCode2().

Second-generation codes are fast to work with, since they are used internally by the NPerm4 class.

Returns
the second-generation permutation code.
NPerm4 regina::NPerm4::inverse ( ) const
inline

Finds the inverse of this permutation.

Returns
the inverse of this permutation.
bool regina::NPerm4::isIdentity ( ) const
inline

Determines if this is the identity permutation.

This is true if and only if each of 0, 1, 2 and 3 is mapped to itself.

Returns
true if and only if this is the identity permutation.
static bool regina::NPerm4::isPermCode ( unsigned char  code)
static

Determines whether the given character is a valid first-generation permutation code.

Valid first-generation codes can be passed to setPermCode() or fromPermCode(), and are returned by getPermCode().

Warning
This routine will incur additional overhead, since NPerm4 now uses second-generation codes internally. See the class notes and the routine isPermCode2() for details.
Parameters
codethe permutation code to test.
Returns
true if and only if the given code is a valid first-generation permutation code.
bool regina::NPerm4::isPermCode2 ( unsigned char  code)
inlinestatic

Determines whether the given character is a valid second-generation permutation code.

Valid second-generation codes can be passed to setPermCode2() or fromPermCode2(), and are returned by getPermCode2().

Second-generation codes are fast to work with, since they are used internally by the NPerm4 class.

Parameters
codethe permutation code to test.
Returns
true if and only if the given code is a valid second-generation permutation code.
bool regina::NPerm4::operator!= ( const NPerm4 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, 2 or 3.

Parameters
otherthe permutation with which to compare this.
Returns
true if and only if this and the given permutation differ.
NPerm4 regina::NPerm4::operator* ( const NPerm4 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.
NPerm4 & regina::NPerm4::operator= ( const NPerm4 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.
bool regina::NPerm4::operator== ( const NPerm4 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, 2 and 3.

Parameters
otherthe permutation with which to compare this.
Returns
true if and only if this and the given permutation are equal.
int regina::NPerm4::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 3 inclusive.
Returns
the image of source.
int regina::NPerm4::orderedS4Index ( ) const
inline

Returns the index of this permutation in the NPerm4::orderedS4 array.

Returns
the index i for which this permutation is equal to NPerm4::orderedS4[i]. This will be between 0 and 23 inclusive.
int regina::NPerm4::orderedSnIndex ( ) const
inline

Returns the index of this permutation in the NPerm4::orderedS4 array.

This is a dimension-agnostic alias for orderedS4Index().

Returns
the index i for which this permutation is equal to NPerm4::orderedS4[i]. This will be between 0 and 23 inclusive.
int regina::NPerm4::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 3 inclusive.
Returns
the preimage of image.
int regina::NPerm4::S4Index ( ) const
inline

Returns the index of this permutation in the NPerm4::S4 array.

Returns
the index i for which this permutation is equal to NPerm4::S4[i]. This will be between 0 and 23 inclusive.
void regina::NPerm4::setPerm ( int  a,
int  b 
)
inline

Sets this permutation to the transposition of a and b.

Note that a and b need not be distinct.

Precondition
a and b are in {0,1,2,3}.
Deprecated:
This routine is largely unnecessary, since NPerm4 objects are tiny and cheap. Just use the assignment operator instead. This routine will eventually be removed entirely in a future version of Regina.
Parameters
athe element to switch with b.
bthe element to switch with a.
void regina::NPerm4::setPerm ( int  a,
int  b,
int  c,
int  d 
)
inline

Sets this permutation to that mapping (0,1,2,3) to (a,b,c,d) respectively.

Precondition
{a,b,c,d} = {0,1,2,3}.
Deprecated:
This routine is largely unnecessary, since NPerm4 objects are tiny and cheap. Just use the assignment operator instead. This routine will eventually be removed entirely in a future version of Regina.
Parameters
athe desired image of 0.
bthe desired image of 1.
cthe desired image of 2.
dthe desired image of 3.
void regina::NPerm4::setPermCode ( unsigned char  code)
inline

Sets this permutation to that represented by the given first-generation permutation code.

Precondition
the given code is a valid first-generation permutation code; see isPermCode() for details.
Warning
This routine will incur additional overhead, since NPerm4 now uses second-generation codes internally. See the class notes and the routine setPermCode2() for details.
Parameters
codethe first-generation code that will determine the new value of this permutation.
void regina::NPerm4::setPermCode2 ( unsigned char  code)
inline

Sets this permutation to that represented by the given second-generation permutation code.

Second-generation codes are fast to work with, since they are used internally by the NPerm4 class.

Precondition
the given code is a valid second-generation permutation code; see isPermCode2() for details.
Parameters
codethe second-generation code that will determine the new value of this permutation.
int regina::NPerm4::sign ( ) const
inline

Determines the sign of this permutation.

Returns
1 if this permutation is even, or -1 if this permutation is odd.
int regina::NPerm4::SnIndex ( ) const
inline

Returns the index of this permutation in the NPerm4::S4 array.

This is a dimension-agnostic alias for S4Index().

Returns
the index i for which this permutation is equal to NPerm4::S4[i]. This will be between 0 and 23 inclusive.
std::string regina::NPerm4::str ( ) const

Returns a string representation of this permutation.

The representation will consist of four adjacent digits representing the images of 0, 1, 2 and 3 respectively. An example of a string representation is 1302.

Returns
a string representation of this permutation.
std::string regina::NPerm4::toString ( ) const
inline

A deprecated alias for str(), which returns a string representation of this permutation.

Deprecated:
This routine has (at long last) been deprecated; use the simpler-to-type str() instead.
Returns
a string representation of this permutation.
std::string regina::NPerm4::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.
std::string regina::NPerm4::trunc3 ( ) const

Returns a string representation of this permutation with only the images of 0, 1 and 2 included.

The resulting string will therefore have length three.

Returns
a truncated string representation of this permutation.

Friends And Related Function Documentation

std::ostream& operator<< ( std::ostream &  out,
const NPerm4 p 
)
friend

Writes a string representation of the given permutation to the given output stream.

The format will be the same as is used by NPerm4::str().

Parameters
outthe output stream to which to write.
pthe permutation to write.
Returns
a reference to out.

Member Data Documentation

const unsigned regina::NPerm4::invS2[2]
static

Contains the inverses of the permutations in the array S2.

Specifically, the inverse of permutation S2[i] is the permutation S2[ invS2[i] ].

Deprecated:
This array is unnecessary, since all elements of S2 are their own inverses. This array will be removed in some future version of Regina.
const unsigned regina::NPerm4::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] ].

Deprecated:
This is identical to the array NPerm3::invS3. This unnecessary copy in NPerm4 will be removed in some future version of Regina.
const unsigned regina::NPerm4::invS4[24]
static

Contains the inverses of the permutations in the array S4.

Specifically, the inverse of permutation S4[i] is the permutation S4[ invS4[i] ].

const unsigned* regina::NPerm4::invSn
static

A dimension-agnostic alias for NPerm4::invS4.

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

const NPerm4 regina::NPerm4::orderedS3[6]
static

Contains all possible permutations of three elements in lexicographical order.

In each permutation, 3 maps to 3.

const NPerm4 regina::NPerm4::orderedS4[24]
static

Contains all possible permutations of four elements in lexicographical order.

const NPerm4* regina::NPerm4::orderedSn
static

A dimension-agnostic alias for NPerm4::orderedS4.

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

const NPerm4 regina::NPerm4::S2[2]
static

Contains all possible permutations of two elements.

In each permutation, 2 maps to 2 and 3 maps to 3.

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 (NPerm4, NPerm5 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.

const NPerm4 regina::NPerm4::S3[6]
static

Contains all possible permutations of three elements.

In each permutation, 3 maps to 3.

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 (NPerm4, NPerm5 and so on), the S3 array stores the same permutations in the same order (but of course using different data types).

Note that the permutations are not necessarily in lexicographical order. For the corresponding inverse array, see NPerm3::invS3.

const NPerm4 regina::NPerm4::S4[24]
static

Contains all possible permutations of four 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 (NPerm4, NPerm5 and so on), the S4 array stores the same permutations in the same order (but of course using different data types).

Note that the permutations are not necessarily in lexicographical order.

const NPerm4* regina::NPerm4::Sn
static

A dimension-agnostic alias for NPerm4::S4.

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

const NPerm4* regina::NPerm4::Sn_1
static

A dimension-agnostic alias for NPerm4::S3.

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 © 1999-2014, 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@debian.org).