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Matthias Koeppe
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sage.combinat: More # optional
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‎src/sage/combinat/species/generating_series.py

Lines changed: 28 additions & 28 deletions
Original file line numberDiff line numberDiff line change
@@ -17,7 +17,7 @@
1717
1818
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
1919
sage: p = SymmetricFunctions(QQ).power() # optional - sage.modules
20-
sage: CIS = CycleIndexSeriesRing(QQ)
20+
sage: CIS = CycleIndexSeriesRing(QQ) # optional - sage.modules
2121
sage: geo1 = CIS(lambda i: p([1])^i) # optional - sage.modules
2222
sage: geo2 = CIS(lambda i: p([2])^(i // 2) if is_even(i) else 0) # optional - sage.modules
2323
sage: s = geo1 * geo2 # optional - sage.modules
@@ -294,7 +294,7 @@ def count(self, t):
294294
295295
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
296296
sage: p = SymmetricFunctions(QQ).power() # optional - sage.modules
297-
sage: CIS = CycleIndexSeriesRing(QQ)
297+
sage: CIS = CycleIndexSeriesRing(QQ) # optional - sage.modules
298298
sage: f = CIS([0, p([1]), 2*p([1,1]), 3*p([2,1])]) # optional - sage.modules
299299
sage: f.count([1]) # optional - sage.modules
300300
1
@@ -314,7 +314,7 @@ def coefficient_cycle_type(self, t):
314314
315315
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
316316
sage: p = SymmetricFunctions(QQ).power() # optional - sage.modules
317-
sage: CIS = CycleIndexSeriesRing(QQ)
317+
sage: CIS = CycleIndexSeriesRing(QQ) # optional - sage.modules
318318
sage: f = CIS([0, p([1]), 2*p([1,1]),3*p([2,1])]) # optional - sage.modules
319319
sage: f.coefficient_cycle_type([1]) # optional - sage.modules
320320
1
@@ -334,9 +334,9 @@ def isotype_generating_series(self):
334334
EXAMPLES::
335335
336336
sage: P = species.PermutationSpecies()
337-
sage: cis = P.cycle_index_series()
338-
sage: f = cis.isotype_generating_series()
339-
sage: f[:10]
337+
sage: cis = P.cycle_index_series() # optional - sage.modules
338+
sage: f = cis.isotype_generating_series() # optional - sage.modules
339+
sage: f[:10] # optional - sage.modules
340340
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
341341
"""
342342
R = self.base_ring()
@@ -352,8 +352,8 @@ def _ogs_gen(self, n, ao):
352352
EXAMPLES::
353353
354354
sage: P = species.PermutationSpecies()
355-
sage: cis = P.cycle_index_series()
356-
sage: [cis._ogs_gen(i, 0) for i in range(10)]
355+
sage: cis = P.cycle_index_series() # optional - sage.modules
356+
sage: [cis._ogs_gen(i, 0) for i in range(10)] # optional - sage.modules
357357
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
358358
"""
359359
if n < ao:
@@ -367,9 +367,9 @@ def generating_series(self):
367367
EXAMPLES::
368368
369369
sage: P = species.PartitionSpecies()
370-
sage: cis = P.cycle_index_series()
371-
sage: f = cis.generating_series()
372-
sage: f[:5]
370+
sage: cis = P.cycle_index_series() # optional - sage.modules
371+
sage: f = cis.generating_series() # optional - sage.modules
372+
sage: f[:5] # optional - sage.modules
373373
[1, 1, 1, 5/6, 5/8]
374374
"""
375375
R = self.base_ring()
@@ -385,8 +385,8 @@ def _egs_gen(self, n, ao):
385385
EXAMPLES::
386386
387387
sage: P = species.PermutationSpecies()
388-
sage: cis = P.cycle_index_series()
389-
sage: [cis._egs_gen(i, 0) for i in range(10)]
388+
sage: cis = P.cycle_index_series() # optional - sage.modules
389+
sage: [cis._egs_gen(i, 0) for i in range(10)] # optional - sage.modules
390390
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
391391
"""
392392
if n < ao:
@@ -409,16 +409,16 @@ def derivative(self, n=1):
409409
410410
The species `E` of sets satisfies the relationship `E' = E`::
411411
412-
sage: E = species.SetSpecies().cycle_index_series()
413-
sage: E[:8] == E.derivative()[:8]
412+
sage: E = species.SetSpecies().cycle_index_series() # optional - sage.modules
413+
sage: E[:8] == E.derivative()[:8] # optional - sage.modules
414414
True
415415
416416
The species `C` of cyclic orderings and the species `L` of linear
417417
orderings satisfy the relationship `C' = L`::
418418
419-
sage: C = species.CycleSpecies().cycle_index_series()
420-
sage: L = species.LinearOrderSpecies().cycle_index_series()
421-
sage: L[:8] == C.derivative()[:8]
419+
sage: C = species.CycleSpecies().cycle_index_series() # optional - sage.modules
420+
sage: L = species.LinearOrderSpecies().cycle_index_series() # optional - sage.modules
421+
sage: L[:8] == C.derivative()[:8] # optional - sage.modules
422422
True
423423
"""
424424
return self.derivative_with_respect_to_p1(n=n)
@@ -439,9 +439,9 @@ def pointing(self):
439439
The species `E^{\bullet}` of "pointed sets" satisfies
440440
`E^{\bullet} = X \cdot E`::
441441
442-
sage: E = species.SetSpecies().cycle_index_series()
443-
sage: X = species.SingletonSpecies().cycle_index_series()
444-
sage: E.pointing()[:8] == (X*E)[:8]
442+
sage: E = species.SetSpecies().cycle_index_series() # optional - sage.modules
443+
sage: X = species.SingletonSpecies().cycle_index_series() # optional - sage.modules
444+
sage: E.pointing()[:8] == (X*E)[:8] # optional - sage.modules
445445
True
446446
"""
447447
X = self.parent()([1], valuation=1)
@@ -463,9 +463,9 @@ def exponential(self):
463463
Let `BT` be the species of binary trees, `BF` the species of binary
464464
forests, and `E` the species of sets. Then we have `BF = E \circ BT`::
465465
466-
sage: BT = species.BinaryTreeSpecies().cycle_index_series()
467-
sage: BF = species.BinaryForestSpecies().cycle_index_series()
468-
sage: BT.exponential().isotype_generating_series()[:8] == BF.isotype_generating_series()[:8]
466+
sage: BT = species.BinaryTreeSpecies().cycle_index_series() # optional - sage.modules
467+
sage: BF = species.BinaryForestSpecies().cycle_index_series() # optional - sage.modules
468+
sage: BT.exponential().isotype_generating_series()[:8] == BF.isotype_generating_series()[:8] # optional - sage.modules
469469
True
470470
"""
471471
base_ring = self.parent().base_ring().base_ring()
@@ -490,10 +490,10 @@ def logarithm(self):
490490
Let `G` be the species of nonempty graphs and `CG` be the species of nonempty connected
491491
graphs. Then `G = E^{+} \circ CG`, so `CG = \Omega \circ G`::
492492
493-
sage: G = species.SimpleGraphSpecies().cycle_index_series() - 1
493+
sage: G = species.SimpleGraphSpecies().cycle_index_series() - 1 # optional - sage.modules
494494
sage: from sage.combinat.species.generating_series import LogarithmCycleIndexSeries
495-
sage: CG = LogarithmCycleIndexSeries()(G)
496-
sage: CG.isotype_generating_series()[0:8]
495+
sage: CG = LogarithmCycleIndexSeries()(G) # optional - sage.modules
496+
sage: CG.isotype_generating_series()[0:8] # optional - sage.modules
497497
[0, 1, 1, 2, 6, 21, 112, 853]
498498
"""
499499
base_ring = self.parent().base_ring().base_ring()
@@ -546,7 +546,7 @@ class CycleIndexSeriesRing(LazySymmetricFunctions):
546546
547547
We test to make sure that caching works::
548548
549-
sage: R is CycleIndexSeriesRing(QQ)
549+
sage: R is CycleIndexSeriesRing(QQ) # optional - sage.modules
550550
True
551551
"""
552552
Element = CycleIndexSeries

‎src/sage/combinat/species/library.py

Lines changed: 6 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -43,7 +43,7 @@ def SimpleGraphSpecies():
4343
sage: S = species.SimpleGraphSpecies()
4444
sage: S.generating_series().counts(10)
4545
[1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736]
46-
sage: S.cycle_index_series()[:5]
46+
sage: S.cycle_index_series()[:5] # optional - sage.modules
4747
[p[],
4848
p[1],
4949
p[1, 1] + p[2],
@@ -94,7 +94,7 @@ def BinaryTreeSpecies():
9494
sage: B = species.BinaryTreeSpecies()
9595
sage: a = B.structures([1,2,3,4,5])[187]; a
9696
2*((5*3)*(4*1))
97-
sage: a.automorphism_group()
97+
sage: a.automorphism_group() # optional - sage.groups
9898
Permutation Group with generators [()]
9999
100100
TESTS::
@@ -121,9 +121,9 @@ def BinaryForestSpecies():
121121
sage: F = species.BinaryForestSpecies()
122122
sage: F.generating_series().counts(10)
123123
[1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353]
124-
sage: F.isotype_generating_series().counts(10)
124+
sage: F.isotype_generating_series().counts(10) # optional - sage.modules
125125
[1, 1, 2, 4, 10, 26, 77, 235, 758, 2504]
126-
sage: F.cycle_index_series()[:7]
126+
sage: F.cycle_index_series()[:7] # optional - sage.modules
127127
[p[],
128128
p[1],
129129
3/2*p[1, 1] + 1/2*p[2],
@@ -134,8 +134,8 @@ def BinaryForestSpecies():
134134
135135
TESTS::
136136
137-
sage: seq = F.isotype_generating_series().counts(10)[1:]
138-
sage: oeis(seq)[0] # optional -- internet
137+
sage: seq = F.isotype_generating_series().counts(10)[1:] # optional - sage.modules
138+
sage: oeis(seq)[0] # optional -- internet # optional - sage.modules
139139
A052854: Number of forests of ordered trees on n total nodes.
140140
"""
141141
B = BinaryTreeSpecies()

‎src/sage/combinat/species/recursive_species.py

Lines changed: 6 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -263,7 +263,7 @@ def _cis(self, series_ring, base_ring):
263263
EXAMPLES::
264264
265265
sage: F = CombinatorialSpecies()
266-
sage: F.cycle_index_series()
266+
sage: F.cycle_index_series() # optional - sage.modules
267267
Uninitialized Lazy Series
268268
"""
269269
if base_ring not in self._cycle_index_series:
@@ -429,17 +429,17 @@ def _add_to_digraph(self, d):
429429
430430
EXAMPLES::
431431
432-
sage: d = DiGraph(multiedges=True)
432+
sage: d = DiGraph(multiedges=True) # optional - sage.graphs
433433
sage: X = species.SingletonSpecies()
434434
sage: B = species.CombinatorialSpecies()
435435
sage: B.define(X+B*B)
436-
sage: B._add_to_digraph(d); d
436+
sage: B._add_to_digraph(d); d # optional - sage.graphs
437437
Multi-digraph on 4 vertices
438438
439439
TESTS::
440440
441441
sage: C = species.CombinatorialSpecies()
442-
sage: C._add_to_digraph(d)
442+
sage: C._add_to_digraph(d) # optional - sage.graphs
443443
Traceback (most recent call last):
444444
...
445445
NotImplementedError
@@ -461,15 +461,15 @@ def _equation(self, var_mapping):
461461
EXAMPLES::
462462
463463
sage: C = species.CombinatorialSpecies()
464-
sage: C.algebraic_equation_system()
464+
sage: C.algebraic_equation_system() # optional - sage.graphs
465465
Traceback (most recent call last):
466466
...
467467
NotImplementedError
468468
469469
::
470470
471471
sage: B = species.BinaryTreeSpecies()
472-
sage: B.algebraic_equation_system()
472+
sage: B.algebraic_equation_system() # optional - sage.graphs
473473
[-node3^2 + node1, -node1 + node3 + (-z)]
474474
"""
475475
try:

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