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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>
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/>.
*/
#ifndef _PERMUTATION_H
#define _PERMUTATION_H
#include "la_traits.h"
#include "vec.h"
#include "polynomial.h"
#include "nonclass.h"
typedef unsigned long long PERM_RANK_TYPE;
//permutations are always numbered from 1; offset is employed when applied to vectors and matrices
namespace LA {
//forward declaration
template <typename T> class CyclePerm;
template <typename T> class Partition;
template <typename T> class CompressedPartition;
template <typename T> class YoungTableaux;
template <typename T, typename R> class WeightPermutation;
template <typename T, typename R> class PermutationAlgebra;
//operator== != < > inherited from NRVec
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) {};
explicit NRPerm(const CyclePerm<T> &rhs, const int n=0);
//specific operations
int size() const {return NRVec_from1<T>::size();};
void identity();
bool is_valid() const; //is it really a permutation
bool is_identity() const;
CompressedPartition<T> cycles() const {return CyclePerm<T>(*this).cycles(size());};
NRPerm inverse() const;
NRPerm reverse() const; //backward order
NRPerm operator&(const NRPerm &rhs) const; //concatenate the permutations this,rhs, renumbering rhs (not commutative)
NRPerm operator|(const NRPerm &rhs) const; //concatenate the permutations rhs,this, renumbering rhs (not commutative)
NRPerm operator*(const NRPerm &q) const; //q is rhs and applied first, this applied second
NRPerm operator*(const CyclePerm<T> &r) const;
NRPerm multiply(const NRPerm<T> &q, bool inverse) const; //multiplication but optionally q inversed
NRPerm conjugate_by(const NRPerm &q, bool reverse=false) const; //q^-1 p q or q p q^-1
NRPerm commutator(const NRPerm &q, bool inverse=false) const; //p^-1 q^-1 p q or q^-1 p^-1 q p
int parity() const; //returns +/- 1
void randomize(void); //uniformly random by Fisher-Yates shuffle
bool next(); //generate next permutation in lex order
PERM_RANK_TYPE generate_all(void (*callback)(const NRPerm<T>&), int parity_select=0); //Algorithm L from Knuth's vol.4, efficient but not in lex order!
PermutationAlgebra<T,T> list_all(int parity_select=0);
PermutationAlgebra<T,T> list_all_lex();
PERM_RANK_TYPE generate_all_multi(void (*callback)(const NRPerm<T>&)); //Algorithm L2 from Knuth's vol.4, for a multiset (repeated numbers, not really permutations)
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 generate_restricted(void (*callback)(const NRPerm<T>&), const NRVec_from1<T> &classes, int restriction_type=0);
PermutationAlgebra<T,T> list_restricted(const NRVec_from1<T> &classes, int restriction_type=0, bool inverted=false); //weight is set to parity (antisymmetrizer) by default
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
NRPerm pow(const int n) const {return power(*this,n);};
};
//this is not a class memeber due to double templating
//it is also possible to use member function permuted of NRVec(_from1)
template <typename T, typename X>
NRVec_from1<X> applypermutation(const NRPerm<T> &p, const NRVec_from1<X> &set, bool inverse=false)
{
#ifdef DEBUG
if(p.size()!=set.size()) laerror("size mismatch in applypermutation");
#endif
NRVec_from1<X> r(set.size());
if(inverse) for(int i=1; i<=p.size(); ++i) r[p[i]] = set[i];
else for(int i=1; i<=p.size(); ++i) r[i] = set[p[i]];
return r;
}
template <typename T, typename R>
class WeightPermutation {
public:
R weight;
NRPerm<T> perm;
int size() const {return perm.size();};
bool is_zero() const {return weight==0;}
bool is_scaledidentity() const {return perm.is_identity();}
bool is_identity() const {return weight==1 && is_scaledidentity();}
bool is_plaindata() const {return false;};
WeightPermutation() : weight(0) {};
WeightPermutation(const R w, const NRPerm<T> &p) : weight(w), perm(p) {};
WeightPermutation(const NRPerm<T> &p) : perm(p) {weight= p.parity();};
void copyonwrite() {perm.copyonwrite();};
WeightPermutation operator&(const WeightPermutation &rhs) const {return WeightPermutation(weight*rhs.weight,perm&rhs.perm);};
WeightPermutation operator|(const WeightPermutation &rhs) const {return WeightPermutation(weight*rhs.weight,perm|rhs.perm);};
WeightPermutation operator*(const WeightPermutation &rhs) const {return WeightPermutation(weight*rhs.weight,perm*rhs.perm);};
WeightPermutation operator*(const R &x) const {return WeightPermutation(weight*x,perm); }
bool operator==(const WeightPermutation &rhs) const {return this->perm == rhs.perm;}; //NOTE for sorting, compares only the permutation not the weight!
bool operator!=(const WeightPermutation &rhs) const {return !(*this==rhs);} //NOTE: compares only the permutation
bool operator>(const WeightPermutation &rhs) const {return this->perm > rhs.perm;};
bool operator<(const WeightPermutation &rhs) const {return this->perm < rhs.perm;};
bool operator>=(const WeightPermutation &rhs) const {return !(*this < rhs);};
bool operator<=(const WeightPermutation &rhs) const {return !(*this > rhs);};
WeightPermutation & operator=(const WeightPermutation &rhs) {weight=rhs.weight; perm=rhs.perm; return *this;};
};
//some necessary traits of the non-scalar class to be able to use LA methods
template<typename T, typename R>
class LA_traits<WeightPermutation<T,R> > {
public:
static bool is_plaindata() {return false;};
static void copyonwrite(WeightPermutation<T,R>& x) {x.perm.copyonwrite();};
typedef typename LA_traits<R>::normtype normtype;
typedef R coefficienttype;
typedef NRPerm<T> elementtype;
static inline bool smaller(const WeightPermutation<T,R>& x, const WeightPermutation<T,R>& y) {return x.perm<y.perm;};
static inline bool bigger(const WeightPermutation<T,R>& x, const WeightPermutation<T,R>& y) {return x.perm>y.perm;};
static R coefficient(const WeightPermutation<T,R>& x){return x.weight;};
static R& coefficient(WeightPermutation<T,R>& x) {return x.weight;};
static typename LA_traits<R>::normtype abscoefficient(const WeightPermutation<T,R>& x){return LA_traits<R>::abs2(x.weight);};
static void clear(WeightPermutation<T,R> *v, int nn) {for(int i=0; i<nn; ++i) {v[i].weight=0; v[i].perm.clear();}}
};
template <typename T, typename R>
std::istream & operator>>(std::istream &s, WeightPermutation<T,R> &x)
{
s>>x.weight>>x.perm;
return s;
}
template <typename T, typename R>
std::ostream & operator<<(std::ostream &s, const WeightPermutation<T,R> &x)
{
s<<x.weight<<' '<<x.perm<<' ';
return s;
}
template <typename T, typename R>
class PermutationAlgebra : public NRVec<WeightPermutation<T,R> >
{
public:
PermutationAlgebra() {};
PermutationAlgebra(int n) : NRVec<WeightPermutation<T,R> >(n) {};
PermutationAlgebra(const NRVec<WeightPermutation<T,R> > &x) : NRVec<WeightPermutation<T,R> >(x) {};
int size() const {return NRVec<WeightPermutation<T,R> >::size();};
void copyonwrite() {NRVec<WeightPermutation<T,R> >::copyonwrite();};
int sort(int direction = 0, int from = 0, int to = -1, int *permut = NULL, bool stable=false) {return NRVec<WeightPermutation<T,R> >::sort(direction,from, to,permut,stable);};
PermutationAlgebra operator&(const PermutationAlgebra &rhs) const;
PermutationAlgebra operator|(const PermutationAlgebra &rhs) const;
PermutationAlgebra operator*(const PermutationAlgebra &rhs) const;
PermutationAlgebra operator+(const PermutationAlgebra &rhs) const;
PermutationAlgebra &operator*=(const R &x) {this->copyonwrite(); for(int i=1; i<=size(); ++i) (*this)[i].weight *= x; return *this;};
PermutationAlgebra operator*(const R &x) const {PermutationAlgebra r(*this); return r*=x;};
void simplify(const typename LA_traits<R>::normtype thr=0) {NRVec_simplify(*this,thr);};
bool operator==(PermutationAlgebra &rhs); //do NOT inherit from NRVec, as the underlying one ignores weights for the simplification; also we have to simplify before comparison
bool is_zero() const {return size()==0;}; //assume it was simplified
bool is_scaled_identity() const {return size()==1 && (*this)[0].is_scaledidentity();}; //assume it was simplified
bool is_identity() const {return size()==1 && (*this)[0].is_identity();}; //assume it was simplified
};
extern PERM_RANK_TYPE factorial(const int n);
extern PERM_RANK_TYPE binom(int n, int k);
extern PERM_RANK_TYPE longpow(PERM_RANK_TYPE x, int i);
//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?
explicit CyclePerm(const NRPerm<T> &rhs);
bool is_valid() const; //is it really a permutation
bool is_identity() const; //no cycles of length > 1
void identity() {this->resize(0);};
CyclePerm inverse() const; //reverse all cycles
int parity() const; //negative if having odd number of even-length cycles
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;}
CompressedPartition<T> cycles(T n = 0) const;
void readfrom(const std::string &line);
CyclePerm operator*(const CyclePerm &q) const; //q is rhs and applied first, this applied second
NRPerm<T> operator*(const NRPerm<T> &r) const;
CyclePerm conjugate_by(const CyclePerm &q) const; //q^-1 p q
PERM_RANK_TYPE order() const; //lcm of cycle lengths
bool operator==(const CyclePerm &rhs) const {return NRPerm<T>(*this) == NRPerm<T>(rhs);}; //cycle representation is not unique, cannot inherit operator== from NRVec
void simplify(bool keep1=false); //remove cycles of size 0 or 1
CyclePerm pow_simple(const int n) const {return CyclePerm(NRPerm<T>(*this).pow(n));}; //do not call power() with our operator*
CyclePerm pow(const int n, const bool keep1=false) const; //a more efficient algorithm
};
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)
{
return (a/gcd(a,b))*b;
}
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);
//compressed partitions stored as #of 1s, #of 2s, etc.
template <typename T>
class CompressedPartition : public NRVec_from1<T> {
public:
CompressedPartition(): NRVec_from1<T>() {};
template<int SIZE> explicit CompressedPartition(const T (&a)[SIZE]) : NRVec_from1<T>(a) {};
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;}
bool is_valid() const {return this->size() == this->sum();}
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, returns +/- 1
};
template <typename T>
std::ostream & operator<<(std::ostream &s, const CompressedPartition<T> &x);
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) {};
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; }
explicit Partition(const YoungTableaux<T> &x); //extract a partition as a shape of Young tableaux
Partition adjoint() const; //also called conjugate partition
PERM_RANK_TYPE Sn_irrep_dim() const;
PERM_RANK_TYPE Un_irrep_dim(const int n) const;
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, returns +/- 1
};
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));
}
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?
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
PERM_RANK_TYPE generate_all_standard(void (*callback)(const YoungTableaux<T>&));
PermutationAlgebra<T,T> young_operator() const; //generate young operator for a standard tableaux
};
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
//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
T irrepdim(T i) const {return chi(i,1);};
T sumirrepdims() const {T s=0; for(T i=1; i<=chi.nrows(); ++i) s+=irrepdim(i); return s;};
};
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);
template<typename T>
const NRVec<T> NRVec<T>::permuted(const NRPerm<int> &p, const bool inverse) const
{
#ifdef DEBUG
if(!p.is_valid()) laerror("invalid permutation of vector");
#endif
int n=p.size();
if(n!=(*this).size()) laerror("incompatible permutation and vector");
#ifdef CUDALA
if(this->getlocation() != cpu || p.getlocation() != cpu ) laerror("permutations can be done only in CPU memory");
#endif
NRVec<T> r(n);
if(inverse) for(int i=1; i<=n; ++i) r[i-1] = v[p[i]-1];
else for(int i=1; i<=n; ++i) r[p[i]-1] = v[i-1];
return r;
}
template<typename T>
NRMat<PERM_RANK_TYPE> Multable(T n);
template<typename T, typename R>
NRMat<R> RegularRepresentation(const PermutationAlgebra<T,R> &a, const NRMat<PERM_RANK_TYPE> &mtable);
template<typename T>
PermutationAlgebra<T,T> general_antisymmetrizer(const NRVec<NRVec_from1<T> > &groups, int restriction_type=0, bool inverted=false);
}//namespace
#endif
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