LA_library/permutation.h

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/*
LA: linear algebra C++ interface library
Copyright (C) 2021 Jiri Pittner <jiri.pittner@jh-inst.cas.cz> or <jiri@pittnerovi.com>
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This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
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#ifndef _PERMUTATION_H
#define _PERMUTATION_H
#include "la_traits.h"
#include "vec.h"
#include "polynomial.h"
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typedef unsigned long long PERM_RANK_TYPE;
//permutations are always numbered from 1; offset is employed when applied to vectors and matrices
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namespace LA {
//forward declaration
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template <typename T> class CyclePerm;
template <typename T> class Partition;
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template <typename T> class CompressedPartition;
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template <typename T> class YoungTableaux;
template <typename T>
class NRPerm : public NRVec_from1<T> {
public:
//basic constructors
NRPerm(): NRVec_from1<T>() {};
template<int SIZE> explicit NRPerm(const T (&a)[SIZE]) : NRVec_from1<T>(a) {};
NRPerm(const int n) : NRVec_from1<T>(n) {};
NRPerm(const NRVec_from1<T> &rhs): NRVec_from1<T>(rhs) {};
NRPerm(const T *a, const int n): NRVec_from1<T>(a, n) {};
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explicit NRPerm(const CyclePerm<T> &rhs, const int n=0);
//specific operations
void identity();
bool is_valid() const; //is it really a permutation
bool is_identity() const;
NRPerm inverse() const;
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NRPerm reverse() const; //backward order
NRPerm operator*(const NRPerm q) const; //q is rhs and applied first, this applied second
NRPerm conjugate_by(const NRPerm q) const; //q^-1 p q
int parity() const;
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void randomize(void); //uniformly random by Fisher-Yates shuffle
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bool next(); //generate next permutation in lex order
PERM_RANK_TYPE generate_all(void (*callback)(const NRPerm<T>&), int parity_select=0); //Algorithm from Knuth's vol.4, efficient but not in lex order!
PERM_RANK_TYPE generate_all2(void (*callback)(const NRPerm<T>&)); //recursive method, also not lexicographic
PERM_RANK_TYPE generate_all_lex(void (*callback)(const NRPerm<T>&)); //generate in lex order using next()
PERM_RANK_TYPE rank() const; //counted from 0 to n!-1
NRVec_from1<T> inversions(const int type, PERM_RANK_TYPE *prank=NULL) const; //inversion tables
explicit NRPerm(const int type, const NRVec_from1<T> &inversions); //compute permutation from inversions
explicit NRPerm(const int n, const PERM_RANK_TYPE rank); //compute permutation from its rank
};
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extern PERM_RANK_TYPE factorial(const int n);
extern PERM_RANK_TYPE binom(int n, int k);
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extern PERM_RANK_TYPE longpow(PERM_RANK_TYPE x, int i);
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//permutations represented in the cycle format
template <typename T>
class CyclePerm : public NRVec_from1<NRVec_from1<T> > {
public:
CyclePerm() : NRVec_from1<NRVec_from1<T> >() {};
template<int SIZE> explicit CyclePerm(const NRVec_from1<T> (&a)[SIZE]) : NRVec_from1<NRVec_from1<T> >(a) {};
//NOTE - how to do it so that direct nested brace initializer would work?
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explicit CyclePerm(const NRPerm<T> &rhs);
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bool is_valid() const; //is it really a permutation
bool is_identity() const; //no cycles of length > 1
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void identity() {this->resize(0);};
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CyclePerm inverse() const; //reverse all cycles
int parity() const; //negative if having odd number of even-length cycles
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T max() const {T m=0; for(int i=1; i<=this->size(); ++i) {T mm= (*this)[i].max(); if(mm>m) m=mm;} return m;}
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CompressedPartition<T> cycles(const T n) const;
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void readfrom(const std::string &line);
CyclePerm operator*(const CyclePerm q) const; //q is rhs and applied first, this applied second
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PERM_RANK_TYPE order() const; //lcm of cycle lengths
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};
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template <typename T>
T gcd(T big, T small)
{
if(big==0)
{
if(small==0) laerror("bad arguments in gcd");
return small;
}
if(small==0) return big;
if(small==1||big==1) return 1;
T help;
if(small>big) {help=big; big=small; small=help;}
do {
help=small;
small= big%small;
big=help;
}
while(small != 0);
return big;
}
template <typename T>
inline T lcm(T a, T b)
{
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return (a/gcd(a,b))*b;
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}
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template <typename T>
std::istream & operator>>(std::istream &s, CyclePerm<T> &x);
template <typename T>
std::ostream & operator<<(std::ostream &s, const CyclePerm<T> &x);
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//compressed partitions stored as #of 1s, #of 2s, etc.
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template <typename T>
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class CompressedPartition : public NRVec_from1<T> {
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public:
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CompressedPartition(): NRVec_from1<T>() {};
template<int SIZE> explicit CompressedPartition(const T (&a)[SIZE]) : NRVec_from1<T>(a) {};
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CompressedPartition(const int n) : NRVec_from1<T>(n) {};
T sum() const {T s=0; for(int i=1; i<=this->size(); ++i) s += i*(*this)[i]; return s;}
T nparts() const {T s=0; for(int i=1; i<=this->size(); ++i) s += (*this)[i]; return s;}
T nclasses() const {T s=0; for(int i=1; i<=this->size(); ++i) if((*this)[i]) ++s; return s;}
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bool is_valid() const {return this->size() == this->sum();}
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explicit CompressedPartition(const Partition<T> &rhs) : NRVec_from1<T>(rhs.size()) {this->clear(); for(int i=1; i<=rhs.size(); ++i) if(!rhs[i]) break; else (*this)[rhs[i]]++; }
PERM_RANK_TYPE Sn_class_size() const;
int parity() const; //of a permutation with given cycle lengths
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};
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template <typename T>
std::ostream & operator<<(std::ostream &s, const CompressedPartition<T> &x);
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template <typename T>
class Partition : public NRVec_from1<T> {
public:
Partition(): NRVec_from1<T>() {};
template<int SIZE> explicit Partition(const T (&a)[SIZE]) : NRVec_from1<T>(a) {};
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Partition(const int n) : NRVec_from1<T>(n) {};
T nparts() const {T s=0; for(int i=1; i<=this->size(); ++i) if((*this)[i]) ++s; return s;}
bool is_valid() const {if(this->size() != this->sum()) return false; for(int i=2; i<=this->size(); ++i) if((*this)[i]>(*this)[i-1]) return false; return true; }
explicit Partition(const CompressedPartition<T> &rhs) : NRVec_from1<T>(rhs.size()) {this->clear(); int ithru=0; for(int i=rhs.size(); i>=1; --i) for(int j=0; j<rhs[i]; ++j) (*this)[++ithru]=i; }
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explicit Partition(const YoungTableaux<T> &x); //extract a partition as a shape of Young tableaux
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Partition adjoint() const; //also called conjugate partition
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PERM_RANK_TYPE Sn_irrep_dim() const;
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PERM_RANK_TYPE Un_irrep_dim(const int n) const;
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PERM_RANK_TYPE generate_all(void (*callback)(const Partition<T>&), int nparts=0); //nparts <0 means at most to -nparts
int parity() const; //of a permutation with given cycle lengths
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};
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template <typename T>
extern T Sn_character(const Partition<T> &irrep, const Partition<T> &cclass);
template <typename T>
inline T Sn_character(const CompressedPartition<T> &irrep, const CompressedPartition<T> &cclass)
{
return Sn_character(Partition<T>(irrep),Partition<T>(cclass));
}
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template <typename T>
class YoungTableaux : public NRVec_from1<NRVec_from1<T> > {
public:
YoungTableaux() : NRVec_from1<NRVec_from1<T> >() {};
explicit YoungTableaux(const Partition<T> &frame);
template<int SIZE> explicit YoungTableaux(const NRVec_from1<T> (&a)[SIZE]) : NRVec_from1<NRVec_from1<T> >(a) {};
//NOTE - how to do it so that direct nested brace initializer would work?
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bool is_valid() const; //check whether its shape forms a partition
int nrows() const {return this->size();}
int ncols() const {return (*this)[1].size();}
bool is_standard() const; //is it filled in standard way (possibly with repeated numbers)
T sum() const; //get back sum of the partition
T max() const; //get back highest number filled in
NRVec_from1<T> yamanouchi() const; //yamanouchi symbol
T character_contribution(int ncyc=0) const; //contribution of filled tableaux to Sn character
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PERM_RANK_TYPE generate_all_standard(void (*callback)(const YoungTableaux<T>&));
PERM_RANK_TYPE young_operator(void (*callback)(const NRPerm<T>&p, const T parity, const PERM_RANK_TYPE nterms)) const; //generate young operator for a standard tableaux
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};
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template <typename T>
std::ostream & operator<<(std::ostream &s, const YoungTableaux<T> &x);
extern PERM_RANK_TYPE partitions(int n, int k= -1); //enumerate partitions to k parts; k== -1 for total # of partitions
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//Sn character table
template <typename T>
class Sn_characters {
public:
T n;
NRVec_from1<CompressedPartition<T> > classes;
NRVec_from1<CompressedPartition<T> > irreps; //can be in different order than classes
NRVec_from1<PERM_RANK_TYPE> classsizes;
NRMat_from1<T> chi; //characters
Sn_characters(const int n0); //compute the table
bool is_valid() const; //check internal consistency
};
template <typename T> class Polynomial; //forward declaration
template <typename T>
class CycleIndex {
public:
NRVec_from1<CompressedPartition<T> > classes;
NRVec_from1<PERM_RANK_TYPE> classsizes;
CycleIndex(const Sn_characters<T> &rhs): classes(rhs.classes),classsizes(rhs.classsizes) {};
bool is_valid() const; //check internal consistency
Polynomial<T> substitute(const Polynomial<T> &p, PERM_RANK_TYPE *denom) const;
};
template <typename T>
extern std::ostream & operator<<(std::ostream &s, const Sn_characters<T> &c);
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}//namespace
#endif