-
-
Notifications
You must be signed in to change notification settings - Fork 405
/
integer_vector.py
1603 lines (1322 loc) · 48.5 KB
/
integer_vector.py
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
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
"""
(Non-negative) Integer vectors
AUTHORS:
* Mike Hansen (2007) - original module
* Nathann Cohen, David Joyner (2009-2010) - Gale-Ryser stuff
* Nathann Cohen, David Joyner (2011) - Gale-Ryser bugfix
* Travis Scrimshaw (2012-05-12) - Updated doc-strings to tell the user of
that the class's name is a misnomer (that they only contains non-negative
entries).
* Federico Poloni (2013) - specialized ``rank()``
* Travis Scrimshaw (2013-02-04) - Refactored to use ``ClonableIntArray``
"""
# ****************************************************************************
# Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>,
# Copyright (C) 2012 Travis Scrimshaw <tscrim@ucdavis.edu>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code 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.
#
# The full text of the GPL is available at:
#
# https://www.gnu.org/licenses/
# ****************************************************************************
from sage.combinat.integer_lists import IntegerListsLex
from itertools import product
from collections.abc import Sequence
import numbers
from sage.structure.parent import Parent
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.list_clone import ClonableArray
from sage.misc.classcall_metaclass import ClasscallMetaclass
from sage.categories.enumerated_sets import EnumeratedSets
from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.rings.infinity import PlusInfinity
from sage.arith.misc import binomial
from sage.rings.integer_ring import ZZ
from sage.rings.semirings.non_negative_integer_semiring import NN
from sage.rings.integer import Integer
def is_gale_ryser(r,s):
r"""
Tests whether the given sequences satisfy the condition
of the Gale-Ryser theorem.
Given a binary matrix `B` of dimension `n\times m`, the
vector of row sums is defined as the vector whose
`i^{\mbox{th}}` component is equal to the sum of the `i^{\mbox{th}}`
row in `A`. The vector of column sums is defined similarly.
If, given a binary matrix, these two vectors are easy to compute,
the Gale-Ryser theorem lets us decide whether, given two
non-negative vectors `r,s`, there exists a binary matrix
whose row/column sums vectors are `r` and `s`.
This functions answers accordingly.
INPUT:
- ``r``, ``s`` -- lists of non-negative integers.
ALGORITHM:
Without loss of generality, we can assume that:
- The two given sequences do not contain any `0` ( which would
correspond to an empty column/row )
- The two given sequences are ordered in decreasing order
(reordering the sequence of row (resp. column) sums amounts to
reordering the rows (resp. columns) themselves in the matrix,
which does not alter the columns (resp. rows) sums.
We can then assume that `r` and `s` are partitions
(see the corresponding class :class:`Partition`)
If `r^*` denote the conjugate of `r`, the Gale-Ryser theorem
asserts that a binary Matrix satisfying the constraints exists
if and only if `s \preceq r^*`, where `\preceq` denotes
the domination order on partitions.
EXAMPLES::
sage: from sage.combinat.integer_vector import is_gale_ryser
sage: is_gale_ryser([4,2,2],[3,3,1,1])
True
sage: is_gale_ryser([4,2,1,1],[3,3,1,1])
True
sage: is_gale_ryser([3,2,1,1],[3,3,1,1])
False
REMARK: In the literature, what we are calling a
Gale-Ryser sequence sometimes goes by the (rather
generic-sounding) term ''realizable sequence''.
"""
# The sequences only contain non-negative integers
if [x for x in r if x < 0] or [x for x in s if x < 0]:
return False
# builds the corresponding partitions, i.e.
# removes the 0 and sorts the sequences
from sage.combinat.partition import Partition
r2 = Partition(sorted([x for x in r if x>0], reverse=True))
s2 = Partition(sorted([x for x in s if x>0], reverse=True))
# If the two sequences only contained zeroes
if len(r2) == 0 and len(s2) == 0:
return True
rstar = Partition(r2).conjugate()
# same number of 1s domination
return len(rstar) <= len(s2) and sum(r2) == sum(s2) and rstar.dominates(s)
def gale_ryser_theorem(p1, p2, algorithm="gale",
*, solver=None, integrality_tolerance=1e-3):
r"""
Returns the binary matrix given by the Gale-Ryser theorem.
The Gale Ryser theorem asserts that if `p_1,p_2` are two
partitions of `n` of respective lengths `k_1,k_2`, then there is
a binary `k_1\times k_2` matrix `M` such that `p_1` is the vector
of row sums and `p_2` is the vector of column sums of `M`, if
and only if the conjugate of `p_2` dominates `p_1`.
INPUT:
- ``p1, p2``-- list of integers representing the vectors
of row/column sums
- ``algorithm`` -- two possible string values:
- ``'ryser'`` implements the construction due to Ryser [Ryser63]_.
- ``'gale'`` (default) implements the construction due to Gale [Gale57]_.
- ``solver`` -- (default: ``None``) Specify a Mixed Integer Linear Programming
(MILP) solver to be used. If set to ``None``, the default one is used. For
more information on MILP solvers and which default solver is used, see
the method
:meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>`
of the class
:class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``integrality_tolerance`` -- parameter for use with MILP solvers over an
inexact base ring; see :meth:`MixedIntegerLinearProgram.get_values`.
OUTPUT:
A binary matrix if it exists, ``None`` otherwise.
Gale's Algorithm:
(Gale [Gale57]_): A matrix satisfying the constraints of its
sums can be defined as the solution of the following
Linear Program, which Sage knows how to solve.
.. MATH::
\forall i&\sum_{j=1}^{k_2} b_{i,j}=p_{1,j}\\
\forall i&\sum_{j=1}^{k_1} b_{j,i}=p_{2,j}\\
&b_{i,j}\mbox{ is a binary variable}
Ryser's Algorithm:
(Ryser [Ryser63]_): The construction of an `m \times n` matrix
`A=A_{r,s}`, due to Ryser, is described as follows. The
construction works if and only if have `s\preceq r^*`.
* Construct the `m \times n` matrix `B` from `r` by defining
the `i`-th row of `B` to be the vector whose first `r_i`
entries are `1`, and the remainder are 0's, `1 \leq i \leq m`.
This maximal matrix `B` with row sum `r` and ones left
justified has column sum `r^{*}`.
* Shift the last `1` in certain rows of `B` to column `n` in
order to achieve the sum `s_n`. Call this `B` again.
* The `1`'s in column `n` are to appear in those
rows in which `A` has the largest row sums, giving
preference to the bottom-most positions in case of ties.
* Note: When this step automatically "fixes" other columns,
one must skip ahead to the first column index
with a wrong sum in the step below.
* Proceed inductively to construct columns `n-1`, ..., `2`, `1`.
Note: when performing the induction on step `k`, we consider
the row sums of the first `k` columns.
* Set `A = B`. Return `A`.
EXAMPLES:
Computing the matrix for `p_1=p_2=2+2+1`::
sage: from sage.combinat.integer_vector import gale_ryser_theorem
sage: p1 = [2,2,1]
sage: p2 = [2,2,1]
sage: print(gale_ryser_theorem(p1, p2)) # not tested
[1 1 0]
[1 0 1]
[0 1 0]
sage: A = gale_ryser_theorem(p1, p2)
sage: rs = [sum(x) for x in A.rows()]
sage: cs = [sum(x) for x in A.columns()]
sage: p1 == rs; p2 == cs
True
True
Or for a non-square matrix with `p_1=3+3+2+1` and `p_2=3+2+2+1+1`,
using Ryser's algorithm::
sage: from sage.combinat.integer_vector import gale_ryser_theorem
sage: p1 = [3,3,1,1]
sage: p2 = [3,3,1,1]
sage: gale_ryser_theorem(p1, p2, algorithm = "ryser")
[1 1 1 0]
[1 1 0 1]
[1 0 0 0]
[0 1 0 0]
sage: p1 = [4,2,2]
sage: p2 = [3,3,1,1]
sage: gale_ryser_theorem(p1, p2, algorithm = "ryser")
[1 1 1 1]
[1 1 0 0]
[1 1 0 0]
sage: p1 = [4,2,2,0]
sage: p2 = [3,3,1,1,0,0]
sage: gale_ryser_theorem(p1, p2, algorithm = "ryser")
[1 1 1 1 0 0]
[1 1 0 0 0 0]
[1 1 0 0 0 0]
[0 0 0 0 0 0]
sage: p1 = [3,3,2,1]
sage: p2 = [3,2,2,1,1]
sage: print(gale_ryser_theorem(p1, p2, algorithm="gale")) # not tested
[1 1 1 0 0]
[1 1 0 0 1]
[1 0 1 0 0]
[0 0 0 1 0]
With `0` in the sequences, and with unordered inputs::
sage: from sage.combinat.integer_vector import gale_ryser_theorem
sage: gale_ryser_theorem([3,3,0,1,1,0], [3,1,3,1,0], algorithm="ryser")
[1 1 1 0 0]
[1 0 1 1 0]
[0 0 0 0 0]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 0 0]
sage: p1 = [3,1,1,1,1]; p2 = [3,2,2,0]
sage: gale_ryser_theorem(p1, p2, algorithm="ryser")
[1 1 1 0]
[1 0 0 0]
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
TESTS:
This test created a random bipartite graph on `n+m` vertices. Its
adjacency matrix is binary, and it is used to create some
"random-looking" sequences which correspond to an existing matrix. The
``gale_ryser_theorem`` is then called on these sequences, and the output
checked for correction.::
sage: def test_algorithm(algorithm, low = 10, high = 50):
....: n,m = randint(low,high), randint(low,high)
....: g = graphs.RandomBipartite(n, m, .3)
....: s1 = sorted(g.degree([(0,i) for i in range(n)]), reverse = True)
....: s2 = sorted(g.degree([(1,i) for i in range(m)]), reverse = True)
....: m = gale_ryser_theorem(s1, s2, algorithm = algorithm)
....: ss1 = sorted(map(lambda x : sum(x) , m.rows()), reverse = True)
....: ss2 = sorted(map(lambda x : sum(x) , m.columns()), reverse = True)
....: if ((ss1 != s1) or (ss2 != s2)):
....: print("Algorithm %s failed with this input:" % algorithm)
....: print(s1, s2)
sage: for algorithm in ["gale", "ryser"]: # long time
....: for i in range(50):
....: test_algorithm(algorithm, 3, 10)
Null matrix::
sage: gale_ryser_theorem([0,0,0],[0,0,0,0], algorithm="gale")
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
sage: gale_ryser_theorem([0,0,0],[0,0,0,0], algorithm="ryser")
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
REFERENCES:
.. [Ryser63] \H. J. Ryser, Combinatorial Mathematics,
Carus Monographs, MAA, 1963.
.. [Gale57] \D. Gale, A theorem on flows in networks, Pacific J. Math.
7(1957)1073-1082.
"""
from sage.matrix.constructor import matrix
if not is_gale_ryser(p1,p2):
return False
if algorithm == "ryser": # ryser's algorithm
from sage.combinat.permutation import Permutation
# Sorts the sequences if they are not, and remembers the permutation
# applied
tmp = sorted(enumerate(p1), reverse=True, key=lambda x:x[1])
r = [x[1] for x in tmp]
r_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()]
m = len(r)
tmp = sorted(enumerate(p2), reverse=True, key=lambda x:x[1])
s = [x[1] for x in tmp]
s_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()]
# This is the partition equivalent to the sliding algorithm
cols = []
for t in reversed(s):
c = [0] * m
i = 0
while t:
k = i + 1
while k < m and r[i] == r[k]:
k += 1
if t >= k - i: # == number rows of the same length
for j in range(i, k):
r[j] -= 1
c[j] = 1
t -= k - i
else: # Remove the t last rows of that length
for j in range(k-t, k):
r[j] -= 1
c[j] = 1
t = 0
i = k
cols.append(c)
# We added columns to the back instead of the front
A0 = matrix(list(reversed(cols))).transpose()
# Applying the permutations to get a matrix satisfying the
# order given by the input
A0 = A0.matrix_from_rows_and_columns(r_permutation, s_permutation)
return A0
elif algorithm == "gale":
from sage.numerical.mip import MixedIntegerLinearProgram
k1, k2=len(p1), len(p2)
p = MixedIntegerLinearProgram(solver=solver)
b = p.new_variable(binary = True)
for (i,c) in enumerate(p1):
p.add_constraint(p.sum([b[i,j] for j in range(k2)]) ==c)
for (i,c) in enumerate(p2):
p.add_constraint(p.sum([b[j,i] for j in range(k1)]) ==c)
p.set_objective(None)
p.solve()
b = p.get_values(b, convert=ZZ, tolerance=integrality_tolerance)
M = [[0]*k2 for i in range(k1)]
for i in range(k1):
for j in range(k2):
M[i][j] = b[i,j]
return matrix(M)
else:
raise ValueError('the only two algorithms available are "gale" and "ryser"')
def _default_function(l, default, i):
"""
EXAMPLES::
sage: from sage.combinat.integer_vector import _default_function
sage: import functools
sage: f = functools.partial(_default_function, [1,2,3], 99)
sage: f(-1)
99
sage: f(0)
1
sage: f(1)
2
sage: f(2)
3
sage: f(3)
99
"""
try:
if i < 0:
return default
return l[i]
except IndexError:
return default
def list2func(l, default=None):
"""
Given a list ``l``, return a function that takes in a value ``i`` and
return ``l[i]``. If default is not ``None``, then the function will
return the default value for out of range ``i``'s.
EXAMPLES::
sage: f = sage.combinat.integer_vector.list2func([1,2,3])
sage: f(0)
1
sage: f(1)
2
sage: f(2)
3
sage: f(3)
Traceback (most recent call last):
...
IndexError: list index out of range
::
sage: f = sage.combinat.integer_vector.list2func([1,2,3], 0)
sage: f(2)
3
sage: f(3)
0
"""
if default is None:
return lambda i: l[i]
else:
from functools import partial
return partial(_default_function, l, default)
class IntegerVector(ClonableArray):
"""
An integer vector.
"""
def check(self):
"""
Check to make sure this is a valid integer vector by making sure
all entries are non-negative.
EXAMPLES::
sage: IV = IntegerVectors()
sage: elt = IV([1,2,1])
sage: elt.check()
Check :trac:`34510`::
sage: IV3 = IntegerVectors(n=3)
sage: IV3([2,2])
Traceback (most recent call last):
...
ValueError: [2, 2] doesn't satisfy correct constraints
sage: IVk3 = IntegerVectors(k=3)
sage: IVk3([2,2])
Traceback (most recent call last):
...
ValueError: [2, 2] doesn't satisfy correct constraints
sage: IV33 = IntegerVectors(n=3, k=3)
sage: IV33([2,2])
Traceback (most recent call last):
...
ValueError: [2, 2] doesn't satisfy correct constraints
"""
if any(x < 0 for x in self):
raise ValueError("all entries must be non-negative")
if self not in self.parent():
raise ValueError(f"{self} doesn't satisfy correct constraints")
def trim(self):
"""
Remove trailing zeros from the integer vector.
EXAMPLES::
sage: IV = IntegerVectors()
sage: IV([5,3,5,1,0,0]).trim()
[5, 3, 5, 1]
sage: IV([5,0,5,1,0]).trim()
[5, 0, 5, 1]
sage: IV([4,3,3]).trim()
[4, 3, 3]
sage: IV([0,0,0]).trim()
[]
sage: IV = IntegerVectors(k=4)
sage: v = IV([4,3,2,0]).trim(); v
[4, 3, 2]
sage: v.parent()
Integer vectors
"""
P = IntegerVectors()
v = list(self)
if all(i == 0 for i in v):
return P.element_class(P, [], check=False)
while not v[-1]:
v = v[:-1]
return P.element_class(P, v, check=False)
def specht_module(self, base_ring=None):
r"""
Return the Specht module corresponding to ``self``.
EXAMPLES::
sage: SM = IntegerVectors()([2,0,1,0,2]).specht_module(QQ)
sage: SM
Specht module of [(0, 0), (0, 1), (2, 0), (4, 0), (4, 1)] over Rational Field
sage: s = SymmetricFunctions(QQ).s()
sage: s(SM.frobenius_image())
s[2, 2, 1]
"""
from sage.combinat.specht_module import SpechtModule
from sage.combinat.symmetric_group_algebra import SymmetricGroupAlgebra
if base_ring is None:
from sage.rings.rational_field import QQ
base_ring = QQ
R = SymmetricGroupAlgebra(base_ring, sum(self))
return SpechtModule(R, self)
def specht_module_dimension(self, base_ring=None):
r"""
Return the dimension of the Specht module corresponding to ``self``.
INPUT:
- ``BR`` -- (default: `\QQ`) the base ring
EXAMPLES::
sage: IntegerVectors()([2,0,1,0,2]).specht_module_dimension()
5
sage: IntegerVectors()([2,0,1,0,2]).specht_module_dimension(GF(2))
5
"""
from sage.combinat.specht_module import specht_module_rank
return specht_module_rank(self, base_ring)
class IntegerVectors(Parent, metaclass=ClasscallMetaclass):
"""
The class of (non-negative) integer vectors.
INPUT:
- ``n`` -- if set to an integer, returns the combinatorial class
of integer vectors whose sum is ``n``; if set to ``None``
(default), no such constraint is defined
- ``k`` -- the length of the vectors; set to ``None`` (default) if
you do not want such a constraint
.. NOTE::
The entries are non-negative integers.
EXAMPLES:
If ``n`` is not specified, it returns the class of all integer vectors::
sage: IntegerVectors()
Integer vectors
sage: [] in IntegerVectors()
True
sage: [1,2,1] in IntegerVectors()
True
sage: [1, 0, 0] in IntegerVectors()
True
Entries are non-negative::
sage: [-1, 2] in IntegerVectors()
False
If ``n`` is specified, then it returns the class of all integer vectors
which sum to ``n``::
sage: IV3 = IntegerVectors(3); IV3
Integer vectors that sum to 3
Note that trailing zeros are ignored so that ``[3, 0]`` does not show
up in the following list (since ``[3]`` does)::
sage: IntegerVectors(3, max_length=2).list()
[[3], [2, 1], [1, 2], [0, 3]]
If ``n`` and ``k`` are both specified, then it returns the class
of integer vectors that sum to ``n`` and are of length ``k``::
sage: IV53 = IntegerVectors(5,3); IV53
Integer vectors of length 3 that sum to 5
sage: IV53.cardinality()
21
sage: IV53.first()
[5, 0, 0]
sage: IV53.last()
[0, 0, 5]
sage: IV53.random_element().parent() is IV53
True
Further examples::
sage: IntegerVectors(-1, 0, min_part = 1).list()
[]
sage: IntegerVectors(-1, 2, min_part = 1).list()
[]
sage: IntegerVectors(0, 0, min_part=1).list()
[[]]
sage: IntegerVectors(3, 0, min_part=1).list()
[]
sage: IntegerVectors(0, 1, min_part=1).list()
[]
sage: IntegerVectors(2, 2, min_part=1).list()
[[1, 1]]
sage: IntegerVectors(2, 3, min_part=1).list()
[]
sage: IntegerVectors(4, 2, min_part=1).list()
[[3, 1], [2, 2], [1, 3]]
::
sage: IntegerVectors(0, 3, outer=[0,0,0]).list()
[[0, 0, 0]]
sage: IntegerVectors(1, 3, outer=[0,0,0]).list()
[]
sage: IntegerVectors(2, 3, outer=[0,2,0]).list()
[[0, 2, 0]]
sage: IntegerVectors(2, 3, outer=[1,2,1]).list()
[[1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1]]
sage: IntegerVectors(2, 3, outer=[1,1,1]).list()
[[1, 1, 0], [1, 0, 1], [0, 1, 1]]
sage: IntegerVectors(2, 5, outer=[1,1,1,1,1]).list()
[[1, 1, 0, 0, 0],
[1, 0, 1, 0, 0],
[1, 0, 0, 1, 0],
[1, 0, 0, 0, 1],
[0, 1, 1, 0, 0],
[0, 1, 0, 1, 0],
[0, 1, 0, 0, 1],
[0, 0, 1, 1, 0],
[0, 0, 1, 0, 1],
[0, 0, 0, 1, 1]]
::
sage: iv = [ IntegerVectors(n,k) for n in range(-2, 7) for k in range(7) ]
sage: all(map(lambda x: x.cardinality() == len(x.list()), iv))
True
sage: essai = [[1,1,1], [2,5,6], [6,5,2]]
sage: iv = [ IntegerVectors(x[0], x[1], max_part = x[2]-1) for x in essai ]
sage: all(map(lambda x: x.cardinality() == len(x.list()), iv))
True
An example showing the same output by using IntegerListsLex::
sage: IntegerVectors(4, max_length=2).list()
[[4], [3, 1], [2, 2], [1, 3], [0, 4]]
sage: list(IntegerListsLex(4, max_length=2))
[[4], [3, 1], [2, 2], [1, 3], [0, 4]]
.. SEEALSO::
:class:`sage.combinat.integer_lists.invlex.IntegerListsLex`
"""
@staticmethod
def __classcall_private__(cls, n=None, k=None, **kwargs):
"""
Choose the correct parent based upon input.
EXAMPLES::
sage: IV1 = IntegerVectors(3, 2)
sage: IV2 = IntegerVectors(3, 2)
sage: IV1 is IV2
True
TESTS::
sage: IV2 = IntegerVectors(3, 2, length=2)
Traceback (most recent call last):
...
ValueError: k and length both specified
:trac:`29524`::
sage: IntegerVectors(3, 3/1)
Traceback (most recent call last):
...
TypeError: 'k' must be an integer or a tuple, got Rational
"""
if 'length' in kwargs:
if k is not None:
raise ValueError("k and length both specified")
k = kwargs.pop('length')
if kwargs:
return IntegerVectorsConstraints(n, k, **kwargs)
if k is None:
if n is None:
return IntegerVectors_all()
return IntegerVectors_n(n)
if n is None:
return IntegerVectors_k(k)
if isinstance(k, numbers.Integral):
return IntegerVectors_nk(n, k)
elif isinstance(k, (tuple, list)):
return IntegerVectors_nnondescents(n, tuple(k))
else:
raise TypeError("'k' must be an integer or a tuple, got {}".format(type(k).__name__))
def __init__(self, category=None):
"""
Initialize ``self``.
EXAMPLES::
sage: IV = IntegerVectors()
sage: TestSuite(IV).run()
"""
if category is None:
category = EnumeratedSets()
Parent.__init__(self, category=category)
def _element_constructor_(self, lst):
"""
Construct an element of ``self`` from ``lst``.
EXAMPLES::
sage: IV = IntegerVectors()
sage: elt = IV([3, 1, 0, 3, 2]); elt
[3, 1, 0, 3, 2]
sage: elt.parent()
Integer vectors
sage: IV9 = IntegerVectors(9)
sage: elt9 = IV9(elt)
sage: elt9.parent()
Integer vectors that sum to 9
"""
return self.element_class(self, lst)
Element = IntegerVector
def __contains__(self, x):
"""
EXAMPLES::
sage: [] in IntegerVectors()
True
sage: [3,2,2,1] in IntegerVectors()
True
"""
if isinstance(x, IntegerVector):
return True
if not isinstance(x, Sequence):
return False
for i in x:
if i not in ZZ:
return False
if i < 0:
return False
return True
class IntegerVectors_all(UniqueRepresentation, IntegerVectors):
"""
Class of all integer vectors.
"""
def __init__(self):
"""
Initialize ``self``.
EXAMPLES::
sage: IV = IntegerVectors()
sage: TestSuite(IV).run()
"""
IntegerVectors.__init__(self, category=InfiniteEnumeratedSets())
def _repr_(self):
"""
EXAMPLES::
sage: IntegerVectors()
Integer vectors
"""
return "Integer vectors"
def __iter__(self):
"""
Iterate over ``self``.
EXAMPLES::
sage: IV = IntegerVectors()
sage: it = IV.__iter__()
sage: [next(it) for x in range(10)]
[[], [1], [2], [2, 0], [1, 1], [0, 2], [3], [3, 0], [2, 1], [1, 2]]
"""
yield self.element_class(self, [])
n = 1
while True:
for k in range(1, n + 1):
for v in integer_vectors_nk_fast_iter(n, k):
yield self.element_class(self, v, check=False)
n += 1
class IntegerVectors_n(UniqueRepresentation, IntegerVectors):
"""
Integer vectors that sum to `n`.
"""
def __init__(self, n):
"""
TESTS::
sage: IV = IntegerVectors(3)
sage: TestSuite(IV).run()
"""
self.n = n
IntegerVectors.__init__(self, category=InfiniteEnumeratedSets())
def _repr_(self):
"""
TESTS::
sage: IV = IntegerVectors(3)
sage: IV
Integer vectors that sum to 3
"""
return "Integer vectors that sum to {}".format(self.n)
def __iter__(self):
"""
Iterate over ``self``.
EXAMPLES::
sage: it = IntegerVectors(3).__iter__()
sage: [next(it) for x in range(10)]
[[3],
[3, 0],
[2, 1],
[1, 2],
[0, 3],
[3, 0, 0],
[2, 1, 0],
[2, 0, 1],
[1, 2, 0],
[1, 1, 1]]
"""
if not self.n:
yield self.element_class(self, [])
k = 1
while True:
for iv in integer_vectors_nk_fast_iter(self.n, k):
yield self.element_class(self, iv, check=False)
k += 1
def __contains__(self, x):
"""
EXAMPLES::
sage: [0] in IntegerVectors(0)
True
sage: [3] in IntegerVectors(3)
True
sage: [3] in IntegerVectors(2)
False
sage: [3,2,2,1] in IntegerVectors(9)
False
sage: [3,2,2,1] in IntegerVectors(8)
True
"""
if not IntegerVectors.__contains__(self, x):
return False
return sum(x) == self.n
class IntegerVectors_k(UniqueRepresentation, IntegerVectors):
"""
Integer vectors of length `k`.
"""
def __init__(self, k):
"""
TESTS::
sage: IV = IntegerVectors(k=2)
sage: TestSuite(IV).run()
"""
self.k = k
IntegerVectors.__init__(self, category=InfiniteEnumeratedSets())
def _repr_(self):
"""
TESTS::
sage: IV = IntegerVectors(k=2)
sage: IV
Integer vectors of length 2
"""
return "Integer vectors of length {}".format(self.k)
def __iter__(self):
"""
Iterate over ``self``.
EXAMPLES::
sage: it = IntegerVectors(k=2).__iter__()
sage: [next(it) for x in range(10)]
[[0, 0],
[1, 0],
[0, 1],
[2, 0],
[1, 1],
[0, 2],
[3, 0],
[2, 1],
[1, 2],
[0, 3]]
"""
n = 0
while True:
for iv in integer_vectors_nk_fast_iter(n, self.k):
yield self.element_class(self, iv, check=False)
n += 1
def __contains__(self, x):
"""
EXAMPLES::
sage: [] in IntegerVectors(k=0)
True
sage: [3] in IntegerVectors(k=1)
True
sage: [3] in IntegerVectors(k=2)
False
sage: [3,2,2,1] in IntegerVectors(k=3)
False
sage: [3,2,2,1] in IntegerVectors(k=4)
True
"""
if not IntegerVectors.__contains__(self, x):
return False
return len(x) == self.k
class IntegerVectors_nk(UniqueRepresentation, IntegerVectors):
"""
Integer vectors of length `k` that sum to `n`.
AUTHORS:
- Martin Albrecht
- Mike Hansen
"""
def __init__(self, n, k):
"""
TESTS::
sage: IV = IntegerVectors(2, 3)
sage: TestSuite(IV).run()
"""
self.n = n
self.k = k
IntegerVectors.__init__(self, category=FiniteEnumeratedSets())
def _list_rec(self, n, k):
"""
Return a list of a exponent tuples of length ``size`` such
that the degree of the associated monomial is `D`.
INPUT:
- ``n`` -- degree (must be 0)