def test_PR_14617():
from sympy.functions.combinatorial.numbers import nT
for n in (0, []):
for k in (-1, 0, 1):
if k == 0:
assert nT(n, k) == 1
else:
assert nT(n, k) == 0
def test_PR_14617():
from sympy.functions.combinatorial.numbers import nT
for n in (0, []):
for k in (-1, 0, 1):
if k == 0:
assert nT(n, k) == 1
else:
assert nT(n, k) == 0
def processes_set_partitions(processes): number_of_partitions=nT(len(processes)) processes_partitions=Partition(processes) for p in spcon.range(number_of_partitions).collect(): print "=======================================================" print "Partition:",processes_partitions + p print "Partition Rank:",(processes_partitions + p).rank
def test_ordered_partitions():
from sympy.functions.combinatorial.numbers import nT
f = ordered_partitions
assert list(f(0, 1)) == [[]]
assert list(f(1, 0)) == [[]]
for i in range(1, 7):
for j in [None] + list(range(1, i)):
assert (sum(1 for p in f(i, j, 1)) == sum(1 for p in f(i, j, 0)) ==
nT(i, j))
def test_ordered_partitions():
from sympy.functions.combinatorial.numbers import nT
f = ordered_partitions
assert list(f(0, 1)) == [[]]
assert list(f(1, 0)) == [[]]
for i in range(1, 7):
for j in [None] + list(range(1, i)):
assert (
sum(1 for p in f(i, j, 1)) ==
sum(1 for p in f(i, j, 0)) ==
nT(i, j))
def test_nC_nP_nT():
from sympy.utilities.iterables import (multiset_permutations,
multiset_combinations,
multiset_partitions, partitions,
subsets, permutations)
from sympy.functions.combinatorial.numbers import (nP, nC, nT, stirling,
_multiset_histogram,
_AOP_product)
from sympy.combinatorics.permutations import Permutation
from sympy.core.numbers import oo
from random import choice
c = string.ascii_lowercase
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nP(s, i)
tot += check
assert len(list(multiset_permutations(s, i))) == check
if u:
assert nP(len(s), i) == check
assert nP(s) == tot
except AssertionError:
print(s, i, 'failed perm test')
raise ValueError()
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nC(s, i)
tot += check
assert len(list(multiset_combinations(s, i))) == check
if u:
assert nC(len(s), i) == check
assert nC(s) == tot
if u:
assert nC(len(s)) == tot
except AssertionError:
print(s, i, 'failed combo test')
raise ValueError()
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(i, j)
tot += check
assert sum(1 for p in partitions(i, j, size=True)
if p[0] == j) == check
assert nT(i) == tot
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(range(i), j)
tot += check
assert len(list(multiset_partitions(list(range(i)), j))) == check
assert nT(range(i)) == tot
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(1, 8):
check = nT(s, i)
tot += check
assert len(list(multiset_partitions(s, i))) == check
if u:
assert nT(range(len(s)), i) == check
if u:
assert nT(range(len(s))) == tot
assert nT(s) == tot
except AssertionError:
print(s, i, 'failed partition test')
raise ValueError()
# tests for Stirling numbers of the first kind that are not tested in the
# above
assert [stirling(9, i, kind=1) for i in range(11)
] == [0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0]
perms = list(permutations(range(4)))
assert [
sum(1 for p in perms if Permutation(p).cycles == i) for i in range(5)
] == [0, 6, 11, 6, 1] == [stirling(4, i, kind=1) for i in range(5)]
# http://oeis.org/A008275
assert [
stirling(n, k, signed=1) for n in range(10) for k in range(1, n + 1)
] == [
1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50, 35, -10, 1, -120, 274,
-225, 85, -15, 1, 720, -1764, 1624, -735, 175, -21, 1, -5040, 13068,
-13132, 6769, -1960, 322, -28, 1, 40320, -109584, 118124, -67284,
22449, -4536, 546, -36, 1
]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
assert [stirling(n, k, kind=1) for n in range(10)
for k in range(n + 1)] == [
1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35,
10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735,
175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0,
40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1
]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
assert [stirling(n, k, kind=2) for n in range(10)
for k in range(n + 1)] == [
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10,
1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0,
1, 127, 966, 1701, 1050, 266, 28, 1, 0, 1, 255, 3025, 7770,
6951, 2646, 462, 36, 1
]
assert stirling(3, 4, kind=1) == stirling(3, 4, kind=1) == 0
raises(ValueError, lambda: stirling(-2, 2))
def delta(p):
if len(p) == 1:
return oo
return min(abs(i[0] - i[1]) for i in subsets(p, 2))
parts = multiset_partitions(range(5), 3)
d = 2
assert (sum(1
for p in parts if all(delta(i) >= d
for i in p)) == stirling(5, 3, d=d) == 7)
# other coverage tests
assert nC('abb', 2) == nC('aab', 2) == 2
assert nP(3, 3, replacement=True) == nP('aabc', 3, replacement=True) == 27
assert nP(3, 4) == 0
assert nP('aabc', 5) == 0
assert nC(4, 2, replacement=True) == nC('abcdd', 2, replacement=True) == \
len(list(multiset_combinations('aabbccdd', 2))) == 10
assert nC('abcdd') == sum(nC('abcdd', i) for i in range(6)) == 24
assert nC(list('abcdd'), 4) == 4
assert nT('aaaa') == nT(4) == len(list(partitions(4))) == 5
assert nT('aaab') == len(list(multiset_partitions('aaab'))) == 7
assert nC('aabb' * 3, 3) == 4 # aaa, bbb, abb, baa
assert dict(_AOP_product((4, 1, 1, 1))) == {
0: 1,
1: 4,
2: 7,
3: 8,
4: 8,
5: 7,
6: 4,
7: 1
}
# the following was the first t that showed a problem in a previous form of
# the function, so it's not as random as it may appear
t = (3, 9, 4, 6, 6, 5, 5, 2, 10, 4)
assert sum(_AOP_product(t)[i] for i in range(55)) == 58212000
raises(ValueError, lambda: _multiset_histogram({1: 'a'}))
def test_nC_nP_nT():
from sympy.utilities.iterables import (
multiset_permutations, multiset_combinations, multiset_partitions,
partitions, subsets, permutations)
from sympy.functions.combinatorial.numbers import (
nP, nC, nT, stirling, _multiset_histogram, _AOP_product)
from sympy.combinatorics.permutations import Permutation
from sympy.core.numbers import oo
from random import choice
c = string.ascii_lowercase
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nP(s, i)
tot += check
assert len(list(multiset_permutations(s, i))) == check
if u:
assert nP(len(s), i) == check
assert nP(s) == tot
except AssertionError:
print(s, i, 'failed perm test')
raise ValueError()
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nC(s, i)
tot += check
assert len(list(multiset_combinations(s, i))) == check
if u:
assert nC(len(s), i) == check
assert nC(s) == tot
if u:
assert nC(len(s)) == tot
except AssertionError:
print(s, i, 'failed combo test')
raise ValueError()
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(i, j)
tot += check
assert sum(1 for p in partitions(i, j, size=True) if p[0] == j) == check
assert nT(i) == tot
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(range(i), j)
tot += check
assert len(list(multiset_partitions(range(i), j))) == check
assert nT(range(i)) == tot
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(1, 8):
check = nT(s, i)
tot += check
assert len(list(multiset_partitions(s, i))) == check
if u:
assert nT(range(len(s)), i) == check
if u:
assert nT(range(len(s))) == tot
assert nT(s) == tot
except AssertionError:
print(s, i, 'failed partition test')
raise ValueError()
# tests for Stirling numbers of the first kind that are not tested in the
# above
assert [stirling(9, i, kind=1) for i in range(11)] == [
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0]
perms = list(permutations(range(4)))
assert [sum(1 for p in perms if Permutation(p).cycles == i)
for i in range(5)] == [0, 6, 11, 6, 1] == [
stirling(4, i, kind=1) for i in range(5)]
# http://oeis.org/A008275
assert [stirling(n, k, signed=1)
for n in range(10) for k in range(1, n + 1)] == [
1, -1,
1, 2, -3,
1, -6, 11, -6,
1, 24, -50, 35, -10,
1, -120, 274, -225, 85, -15,
1, 720, -1764, 1624, -735, 175, -21,
1, -5040, 13068, -13132, 6769, -1960, 322, -28,
1, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1]
# http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
assert [stirling(n, k, kind=1)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 2, 3, 1,
0, 6, 11, 6, 1,
0, 24, 50, 35, 10, 1,
0, 120, 274, 225, 85, 15, 1,
0, 720, 1764, 1624, 735, 175, 21, 1,
0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1,
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1]
# http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
assert [stirling(n, k, kind=2)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 1, 3, 1,
0, 1, 7, 6, 1,
0, 1, 15, 25, 10, 1,
0, 1, 31, 90, 65, 15, 1,
0, 1, 63, 301, 350, 140, 21, 1,
0, 1, 127, 966, 1701, 1050, 266, 28, 1,
0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1]
assert stirling(3, 4, kind=1) == stirling(3, 4, kind=1) == 0
raises(ValueError, lambda: stirling(-2, 2))
def delta(p):
if len(p) == 1:
return oo
return min(abs(i[0] - i[1]) for i in subsets(p, 2))
parts = multiset_partitions(range(5), 3)
d = 2
assert (sum(1 for p in parts if all(delta(i) >= d for i in p)) ==
stirling(5, 3, d=d) == 7)
# other coverage tests
assert nC('abb', 2) == nC('aab', 2) == 2
assert nP(3, 3, replacement=True) == nP('aabc', 3, replacement=True) == 27
assert nP(3, 4) == 0
assert nP('aabc', 5) == 0
assert nC(4, 2, replacement=True) == nC('abcdd', 2, replacement=True) == \
len(list(multiset_combinations('aabbccdd', 2))) == 10
assert nC('abcdd') == sum(nC('abcdd', i) for i in range(6)) == 24
assert nC(list('abcdd'), 4) == 4
assert nT('aaaa') == nT(4) == len(list(partitions(4))) == 5
assert nT('aaab') == len(list(multiset_partitions('aaab'))) == 7
assert nC('aabb'*3, 3) == 4 # aaa, bbb, abb, baa
assert dict(_AOP_product((4,1,1,1))) == {
0: 1, 1: 4, 2: 7, 3: 8, 4: 8, 5: 7, 6: 4, 7: 1}
# the following was the first t that showed a problem in a previous form of
# the function, so it's not as random as it may appear
t = (3, 9, 4, 6, 6, 5, 5, 2, 10, 4)
assert sum(_AOP_product(t)[i] for i in range(55)) == 58212000
raises(ValueError, lambda: _multiset_histogram({1:'a'}))
