Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

658

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

from __future__ import print_function, division 

 

from sympy.core import Basic, Dict, sympify 

from sympy.core.compatibility import as_int, default_sort_key, range 

from sympy.functions.combinatorial.numbers import bell 

from sympy.matrices import zeros 

from sympy.sets.sets import FiniteSet 

from sympy.utilities.iterables import has_dups, flatten, group 

 

from collections import defaultdict 

 

 

class Partition(FiniteSet): 

""" 

This class represents an abstract partition. 

 

A partition is a set of disjoint sets whose union equals a given set. 

 

See Also 

======== 

 

sympy.utilities.iterables.partitions, 

sympy.utilities.iterables.multiset_partitions 

""" 

 

_rank = None 

_partition = None 

 

def __new__(cls, *partition): 

""" 

Generates a new partition object. 

 

This method also verifies if the arguments passed are 

valid and raises a ValueError if they are not. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> a = Partition([1, 2], [3]) 

>>> a 

{{3}, {1, 2}} 

>>> a.partition 

[[1, 2], [3]] 

>>> len(a) 

2 

>>> a.members 

(1, 2, 3) 

 

""" 

args = partition 

if not all(isinstance(part, (list, FiniteSet)) for part in args): 

raise ValueError( 

"Each argument to Partition should be a list or a FiniteSet") 

 

# sort so we have a canonical reference for RGS 

partition = sorted(sum(partition, []), key=default_sort_key) 

if has_dups(partition): 

raise ValueError("Partition contained duplicated elements.") 

 

obj = FiniteSet.__new__(cls, *[FiniteSet(*x) for x in args]) 

obj.members = tuple(partition) 

obj.size = len(partition) 

return obj 

 

def sort_key(self, order=None): 

"""Return a canonical key that can be used for sorting. 

 

Ordering is based on the size and sorted elements of the partition 

and ties are broken with the rank. 

 

Examples 

======== 

 

>>> from sympy.utilities.iterables import default_sort_key 

>>> from sympy.combinatorics.partitions import Partition 

>>> from sympy.abc import x 

>>> a = Partition([1, 2]) 

>>> b = Partition([3, 4]) 

>>> c = Partition([1, x]) 

>>> d = Partition(list(range(4))) 

>>> l = [d, b, a + 1, a, c] 

>>> l.sort(key=default_sort_key); l 

[{{1, 2}}, {{1}, {2}}, {{1, x}}, {{3, 4}}, {{0, 1, 2, 3}}] 

""" 

if order is None: 

members = self.members 

else: 

members = tuple(sorted(self.members, 

key=lambda w: default_sort_key(w, order))) 

return list(map(default_sort_key, (self.size, members, self.rank))) 

 

@property 

def partition(self): 

"""Return partition as a sorted list of lists. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> Partition([1], [2, 3]).partition 

[[1], [2, 3]] 

""" 

if self._partition is None: 

self._partition = sorted([sorted(p, key=default_sort_key) 

for p in self.args]) 

return self._partition 

 

def __add__(self, other): 

""" 

Return permutation whose rank is ``other`` greater than current rank, 

(mod the maximum rank for the set). 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> a = Partition([1, 2], [3]) 

>>> a.rank 

1 

>>> (a + 1).rank 

2 

>>> (a + 100).rank 

1 

""" 

other = as_int(other) 

offset = self.rank + other 

result = RGS_unrank((offset) % 

RGS_enum(self.size), 

self.size) 

return Partition.from_rgs(result, self.members) 

 

def __sub__(self, other): 

""" 

Return permutation whose rank is ``other`` less than current rank, 

(mod the maximum rank for the set). 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> a = Partition([1, 2], [3]) 

>>> a.rank 

1 

>>> (a - 1).rank 

0 

>>> (a - 100).rank 

1 

""" 

return self.__add__(-other) 

 

def __le__(self, other): 

""" 

Checks if a partition is less than or equal to 

the other based on rank. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> a = Partition([1, 2], [3, 4, 5]) 

>>> b = Partition([1], [2, 3], [4], [5]) 

>>> a.rank, b.rank 

(9, 34) 

>>> a <= a 

True 

>>> a <= b 

True 

""" 

return self.sort_key() <= sympify(other).sort_key() 

 

def __lt__(self, other): 

""" 

Checks if a partition is less than the other. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> a = Partition([1, 2], [3, 4, 5]) 

>>> b = Partition([1], [2, 3], [4], [5]) 

>>> a.rank, b.rank 

(9, 34) 

>>> a < b 

True 

""" 

return self.sort_key() < sympify(other).sort_key() 

 

@property 

def rank(self): 

""" 

Gets the rank of a partition. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> a = Partition([1, 2], [3], [4, 5]) 

>>> a.rank 

13 

""" 

if self._rank is not None: 

return self._rank 

self._rank = RGS_rank(self.RGS) 

return self._rank 

 

@property 

def RGS(self): 

""" 

Returns the "restricted growth string" of the partition. 

 

The RGS is returned as a list of indices, L, where L[i] indicates 

the block in which element i appears. For example, in a partition 

of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is 

[1, 1, 0]: "a" is in block 1, "b" is in block 1 and "c" is in block 0. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> a = Partition([1, 2], [3], [4, 5]) 

>>> a.members 

(1, 2, 3, 4, 5) 

>>> a.RGS 

(0, 0, 1, 2, 2) 

>>> a + 1 

{{3}, {4}, {5}, {1, 2}} 

>>> _.RGS 

(0, 0, 1, 2, 3) 

""" 

rgs = {} 

partition = self.partition 

for i, part in enumerate(partition): 

for j in part: 

rgs[j] = i 

return tuple([rgs[i] for i in sorted( 

[i for p in partition for i in p], key=default_sort_key)]) 

 

@classmethod 

def from_rgs(self, rgs, elements): 

""" 

Creates a set partition from a restricted growth string. 

 

The indices given in rgs are assumed to be the index 

of the element as given in elements *as provided* (the 

elements are not sorted by this routine). Block numbering 

starts from 0. If any block was not referenced in ``rgs`` 

an error will be raised. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import Partition 

>>> Partition.from_rgs([0, 1, 2, 0, 1], list('abcde')) 

{{c}, {a, d}, {b, e}} 

>>> Partition.from_rgs([0, 1, 2, 0, 1], list('cbead')) 

{{e}, {a, c}, {b, d}} 

>>> a = Partition([1, 4], [2], [3, 5]) 

>>> Partition.from_rgs(a.RGS, a.members) 

{{2}, {1, 4}, {3, 5}} 

""" 

if len(rgs) != len(elements): 

raise ValueError('mismatch in rgs and element lengths') 

max_elem = max(rgs) + 1 

partition = [[] for i in range(max_elem)] 

j = 0 

for i in rgs: 

partition[i].append(elements[j]) 

j += 1 

if not all(p for p in partition): 

raise ValueError('some blocks of the partition were empty.') 

return Partition(*partition) 

 

 

class IntegerPartition(Basic): 

""" 

This class represents an integer partition. 

 

In number theory and combinatorics, a partition of a positive integer, 

``n``, also called an integer partition, is a way of writing ``n`` as a 

list of positive integers that sum to n. Two partitions that differ only 

in the order of summands are considered to be the same partition; if order 

matters then the partitions are referred to as compositions. For example, 

4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1]; 

the compositions [1, 2, 1] and [1, 1, 2] are the same as partition 

[2, 1, 1]. 

 

See Also 

======== 

 

sympy.utilities.iterables.partitions, 

sympy.utilities.iterables.multiset_partitions 

 

Reference: http://en.wikipedia.org/wiki/Partition_%28number_theory%29 

""" 

 

_dict = None 

_keys = None 

 

def __new__(cls, partition, integer=None): 

""" 

Generates a new IntegerPartition object from a list or dictionary. 

 

The partition can be given as a list of positive integers or a 

dictionary of (integer, multiplicity) items. If the partition is 

preceeded by an integer an error will be raised if the partition 

does not sum to that given integer. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> a = IntegerPartition([5, 4, 3, 1, 1]) 

>>> a 

IntegerPartition(14, (5, 4, 3, 1, 1)) 

>>> print(a) 

[5, 4, 3, 1, 1] 

>>> IntegerPartition({1:3, 2:1}) 

IntegerPartition(5, (2, 1, 1, 1)) 

 

If the value that the partion should sum to is given first, a check 

will be made to see n error will be raised if there is a discrepancy: 

 

>>> IntegerPartition(10, [5, 4, 3, 1]) 

Traceback (most recent call last): 

... 

ValueError: The partition is not valid 

 

""" 

if integer is not None: 

integer, partition = partition, integer 

if isinstance(partition, (dict, Dict)): 

_ = [] 

for k, v in sorted(list(partition.items()), reverse=True): 

if not v: 

continue 

k, v = as_int(k), as_int(v) 

_.extend([k]*v) 

partition = tuple(_) 

else: 

partition = tuple(sorted(map(as_int, partition), reverse=True)) 

sum_ok = False 

if integer is None: 

integer = sum(partition) 

sum_ok = True 

else: 

integer = as_int(integer) 

 

if not sum_ok and sum(partition) != integer: 

raise ValueError("Partition did not add to %s" % integer) 

if any(i < 1 for i in partition): 

raise ValueError("The summands must all be positive.") 

 

obj = Basic.__new__(cls, integer, partition) 

obj.partition = list(partition) 

obj.integer = integer 

return obj 

 

def prev_lex(self): 

"""Return the previous partition of the integer, n, in lexical order, 

wrapping around to [1, ..., 1] if the partition is [n]. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> p = IntegerPartition([4]) 

>>> print(p.prev_lex()) 

[3, 1] 

>>> p.partition > p.prev_lex().partition 

True 

""" 

d = defaultdict(int) 

d.update(self.as_dict()) 

keys = self._keys 

if keys == [1]: 

return IntegerPartition({self.integer: 1}) 

if keys[-1] != 1: 

d[keys[-1]] -= 1 

if keys[-1] == 2: 

d[1] = 2 

else: 

d[keys[-1] - 1] = d[1] = 1 

else: 

d[keys[-2]] -= 1 

left = d[1] + keys[-2] 

new = keys[-2] 

d[1] = 0 

while left: 

new -= 1 

if left - new >= 0: 

d[new] += left//new 

left -= d[new]*new 

return IntegerPartition(self.integer, d) 

 

def next_lex(self): 

"""Return the next partition of the integer, n, in lexical order, 

wrapping around to [n] if the partition is [1, ..., 1]. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> p = IntegerPartition([3, 1]) 

>>> print(p.next_lex()) 

[4] 

>>> p.partition < p.next_lex().partition 

True 

""" 

d = defaultdict(int) 

d.update(self.as_dict()) 

key = self._keys 

a = key[-1] 

if a == self.integer: 

d.clear() 

d[1] = self.integer 

elif a == 1: 

if d[a] > 1: 

d[a + 1] += 1 

d[a] -= 2 

else: 

b = key[-2] 

d[b + 1] += 1 

d[1] = (d[b] - 1)*b 

d[b] = 0 

else: 

if d[a] > 1: 

if len(key) == 1: 

d.clear() 

d[a + 1] = 1 

d[1] = self.integer - a - 1 

else: 

a1 = a + 1 

d[a1] += 1 

d[1] = d[a]*a - a1 

d[a] = 0 

else: 

b = key[-2] 

b1 = b + 1 

d[b1] += 1 

need = d[b]*b + d[a]*a - b1 

d[a] = d[b] = 0 

d[1] = need 

return IntegerPartition(self.integer, d) 

 

def as_dict(self): 

"""Return the partition as a dictionary whose keys are the 

partition integers and the values are the multiplicity of that 

integer. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> IntegerPartition([1]*3 + [2] + [3]*4).as_dict() 

{1: 3, 2: 1, 3: 4} 

""" 

if self._dict is None: 

groups = group(self.partition, multiple=False) 

self._keys = [g[0] for g in groups] 

self._dict = dict(groups) 

return self._dict 

 

@property 

def conjugate(self): 

""" 

Computes the conjugate partition of itself. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> a = IntegerPartition([6, 3, 3, 2, 1]) 

>>> a.conjugate 

[5, 4, 3, 1, 1, 1] 

""" 

j = 1 

temp_arr = list(self.partition) + [0] 

k = temp_arr[0] 

b = [0]*k 

while k > 0: 

while k > temp_arr[j]: 

b[k - 1] = j 

k -= 1 

j += 1 

return b 

 

def __lt__(self, other): 

"""Return True if self is less than other when the partition 

is listed from smallest to biggest. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> a = IntegerPartition([3, 1]) 

>>> a < a 

False 

>>> b = a.next_lex() 

>>> a < b 

True 

>>> a == b 

False 

""" 

return list(reversed(self.partition)) < list(reversed(other.partition)) 

 

def __le__(self, other): 

"""Return True if self is less than other when the partition 

is listed from smallest to biggest. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> a = IntegerPartition([4]) 

>>> a <= a 

True 

""" 

return list(reversed(self.partition)) <= list(reversed(other.partition)) 

 

def as_ferrers(self, char='#'): 

""" 

Prints the ferrer diagram of a partition. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import IntegerPartition 

>>> print(IntegerPartition([1, 1, 5]).as_ferrers()) 

##### 

# 

# 

""" 

return "\n".join([char*i for i in self.partition]) 

 

def __str__(self): 

return str(list(self.partition)) 

 

 

def random_integer_partition(n, seed=None): 

""" 

Generates a random integer partition summing to ``n`` as a list 

of reverse-sorted integers. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import random_integer_partition 

 

For the following, a seed is given so a known value can be shown; in 

practice, the seed would not be given. 

 

>>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1]) 

[85, 12, 2, 1] 

>>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1]) 

[5, 3, 1, 1] 

>>> random_integer_partition(1) 

[1] 

""" 

from sympy.utilities.randtest import _randint 

 

n = as_int(n) 

if n < 1: 

raise ValueError('n must be a positive integer') 

 

randint = _randint(seed) 

 

partition = [] 

while (n > 0): 

k = randint(1, n) 

mult = randint(1, n//k) 

partition.append((k, mult)) 

n -= k*mult 

partition.sort(reverse=True) 

partition = flatten([[k]*m for k, m in partition]) 

return partition 

 

 

def RGS_generalized(m): 

""" 

Computes the m + 1 generalized unrestricted growth strings 

and returns them as rows in matrix. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import RGS_generalized 

>>> RGS_generalized(6) 

Matrix([ 

[ 1, 1, 1, 1, 1, 1, 1], 

[ 1, 2, 3, 4, 5, 6, 0], 

[ 2, 5, 10, 17, 26, 0, 0], 

[ 5, 15, 37, 77, 0, 0, 0], 

[ 15, 52, 151, 0, 0, 0, 0], 

[ 52, 203, 0, 0, 0, 0, 0], 

[203, 0, 0, 0, 0, 0, 0]]) 

""" 

d = zeros(m + 1) 

for i in range(0, m + 1): 

d[0, i] = 1 

 

for i in range(1, m + 1): 

for j in range(m): 

if j <= m - i: 

d[i, j] = j * d[i - 1, j] + d[i - 1, j + 1] 

else: 

d[i, j] = 0 

return d 

 

 

def RGS_enum(m): 

""" 

RGS_enum computes the total number of restricted growth strings 

possible for a superset of size m. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import RGS_enum 

>>> from sympy.combinatorics.partitions import Partition 

>>> RGS_enum(4) 

15 

>>> RGS_enum(5) 

52 

>>> RGS_enum(6) 

203 

 

We can check that the enumeration is correct by actually generating 

the partitions. Here, the 15 partitions of 4 items are generated: 

 

>>> a = Partition(list(range(4))) 

>>> s = set() 

>>> for i in range(20): 

... s.add(a) 

... a += 1 

... 

>>> assert len(s) == 15 

 

""" 

if (m < 1): 

return 0 

elif (m == 1): 

return 1 

else: 

return bell(m) 

 

 

def RGS_unrank(rank, m): 

""" 

Gives the unranked restricted growth string for a given 

superset size. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import RGS_unrank 

>>> RGS_unrank(14, 4) 

[0, 1, 2, 3] 

>>> RGS_unrank(0, 4) 

[0, 0, 0, 0] 

""" 

if m < 1: 

raise ValueError("The superset size must be >= 1") 

if rank < 0 or RGS_enum(m) <= rank: 

raise ValueError("Invalid arguments") 

 

L = [1] * (m + 1) 

j = 1 

D = RGS_generalized(m) 

for i in range(2, m + 1): 

v = D[m - i, j] 

cr = j*v 

if cr <= rank: 

L[i] = j + 1 

rank -= cr 

j += 1 

else: 

L[i] = int(rank / v + 1) 

rank %= v 

return [x - 1 for x in L[1:]] 

 

 

def RGS_rank(rgs): 

""" 

Computes the rank of a restricted growth string. 

 

Examples 

======== 

 

>>> from sympy.combinatorics.partitions import RGS_rank, RGS_unrank 

>>> RGS_rank([0, 1, 2, 1, 3]) 

42 

>>> RGS_rank(RGS_unrank(4, 7)) 

4 

""" 

rgs_size = len(rgs) 

rank = 0 

D = RGS_generalized(rgs_size) 

for i in range(1, rgs_size): 

n = len(rgs[(i + 1):]) 

m = max(rgs[0:i]) 

rank += D[n, m + 1] * rgs[i] 

return rank