Regina Calculation Engine
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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... | |
NPerm4 & | operator= (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 NPerm4 * | Sn |
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 NPerm4 * | orderedSn |
A dimension-agnostic alias for NPerm4::orderedS4. More... | |
static const NPerm4 | S3 [6] |
Contains all possible permutations of three elements. More... | |
static const NPerm4 * | Sn_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... | |
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).
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Creates the identity permutation.
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Creates the transposition of a and b.
Note that a and b need not be distinct.
a | the element to switch with b. |
b | the element to switch with a. |
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Creates a permutation mapping (0,1,2,3) to (a,b,c,d) respectively.
a | the desired image of 0. |
b | the desired image of 1. |
c | the desired image of 2. |
d | the desired image of 3. |
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Creates a permutation mapping i to image[i] for each i = 0,1,2,3.
image | the 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.
a0 | the desired preimage of a1. |
b0 | the desired preimage of b1. |
c0 | the desired preimage of c1. |
d0 | the desired preimage of d1. |
a1 | the desired image of a0. |
b1 | the desired image of b0. |
c1 | the desired image of c0. |
d1 | the desired image of d0. |
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Creates a permutation that is a clone of the given permutation.
cloneMe | the permutation to clone. |
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Lexicographically compares the images of (0,1,2,3) under this and the given permutation.
other | the permutation with which to compare this. |
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inlinestatic |
Creates a permutation from the given first-generation permutation code.
code | the first-generation code for the new permutation. |
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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.
code | the second-generation code for the new permutation. |
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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().
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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.
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Finds the inverse of this permutation.
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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.
true
if and only if this is the identity permutation.
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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().
code | the permutation code to test. |
true
if and only if the given code is a valid first-generation permutation code.
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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.
code | the permutation code to test. |
true
if and only if the given code is a valid second-generation permutation code.
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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.
other | the permutation with which to compare this. |
true
if and only if this and the given permutation differ. 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]]
.
q | the permutation with which to compose this. |
Sets this permutation to be equal to the given permutation.
cloneMe | the permutation whose value will be assigned to this permutation. |
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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.
other | the permutation with which to compare this. |
true
if and only if this and the given permutation are equal.
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Determines the image of the given integer under this permutation.
source | the integer whose image we wish to find. This should be between 0 and 3 inclusive. |
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Returns the index of this permutation in the NPerm4::orderedS4 array.
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Returns the index of this permutation in the NPerm4::orderedS4 array.
This is a dimension-agnostic alias for orderedS4Index().
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Determines the preimage of the given integer under this permutation.
image | the integer whose preimage we wish to find. This should be between 0 and 3 inclusive. |
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Returns the index of this permutation in the NPerm4::S4 array.
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Sets this permutation to the transposition of a and b.
Note that a and b need not be distinct.
a | the element to switch with b. |
b | the element to switch with a. |
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Sets this permutation to that mapping (0,1,2,3) to (a,b,c,d) respectively.
a | the desired image of 0. |
b | the desired image of 1. |
c | the desired image of 2. |
d | the desired image of 3. |
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Sets this permutation to that represented by the given first-generation permutation code.
code | the first-generation code that will determine the new value of this permutation. |
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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.
code | the second-generation code that will determine the new value of this permutation. |
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Determines the sign of this permutation.
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Returns the index of this permutation in the NPerm4::S4 array.
This is a dimension-agnostic alias for S4Index().
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
.
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A deprecated alias for str(), which 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.
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.
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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().
out | the output stream to which to write. |
p | the permutation to write. |
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Contains the inverses of the permutations in the array S2.
Specifically, the inverse of permutation S2[i]
is the permutation S2[ invS2[i] ]
.
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Contains the inverses of the permutations in the array S3.
Specifically, the inverse of permutation S3[i]
is the permutation S3[ invS3[i] ]
.
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Contains the inverses of the permutations in the array S4.
Specifically, the inverse of permutation S4[i]
is the permutation S4[ invS4[i] ]
.
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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.
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Contains all possible permutations of three elements in lexicographical order.
In each permutation, 3 maps to 3.
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Contains all possible permutations of four elements in lexicographical order.
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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.
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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.
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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.
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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.
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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.
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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).