def testIsotypePartition(tensor):
"""Test that the components of tensor add up to tensor,
that projecting them again the same way leaves them unchanged,
and that projecting them differently leaves them zero"""
#print("---")
total = np.zeros_like(tensor, dtype="float64")
for pp in partitions(tensor.ndim):
p = IntegerPartition(pp).partition
t=project(tensor, p)
#print (p)
#print(";")
#print(t)
#print(t-project(t,p))
#print(project(t,p))
assert np.allclose(t,project(t,p))
#print(tensor,total,t)
total += t
for qq in partitions(tensor.ndim):
q=IntegerPartition(qq).partition
if q!=p:
#print(p,q)
#print(project(t,q))
assert np.allclose(0,project(t,q))
#print(".")
#print(tensor)
#print(total)
assert np.allclose(tensor,total)
print("test ok")
def test_partitions():
assert [p.copy() for p in partitions(6, k=2)] == [
{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=3)] == [
{3: 2}, {1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2},
{1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=2, m=2)] == []
assert [p.copy() for p in partitions(8, k=4, m=3)] == [
{4: 2}, {1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}] == [
i.copy() for i in partitions(8, k=4, m=3) if all(k <= 4 for k in i)
and sum(i.values()) <=3]
assert [p.copy() for p in partitions(S(3), m=2)] == [
{3: 1}, {1: 1, 2: 1}]
assert [i.copy() for i in partitions(4, k=3)] == [
{1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}] == [
i.copy() for i in partitions(4) if all(k <= 3 for k in i)]
raises(ValueError, lambda: list(partitions(3, 0)))
# Consistency check on output of _partitions and RGS_unrank.
# This provides a sanity test on both routines. Also verifies that
# the total number of partitions is the same in each case.
# (from pkrathmann2)
for n in range(2, 6):
i = 0
for m, q in _set_partitions(n):
assert q == RGS_unrank(i, n)
i = i+1
assert i == RGS_enum(n)
def test_partitions():
assert [p.copy() for p in partitions(6, k=2)] == [
{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=3)] == [
{3: 2}, {1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2},
{1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=2, m=2)] == []
assert [p.copy() for p in partitions(8, k=4, m=3)] == [
{4: 2}, {1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}] == [
i.copy() for i in partitions(8, k=4, m=3) if all(k <= 4 for k in i)
and sum(i.values()) <=3]
assert [p.copy() for p in partitions(S(3), m=2)] == [
{3: 1}, {1: 1, 2: 1}]
assert [i.copy() for i in partitions(4, k=3)] == [
{1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}] == [
i.copy() for i in partitions(4) if all(k <= 3 for k in i)]
raises(ValueError, lambda: list(partitions(3, 0)))
# Consistency check on output of _partitions and RGS_unrank.
# This provides a sanity test on both routines. Also verifies that
# the total number of partitions is the same in each case.
# (from pkrathmann2)
for n in range(2, 6):
i = 0
for m, q in _set_partitions(n):
assert q == RGS_unrank(i, n)
i = i+1
assert i == RGS_enum(n)
def test_partitions():
assert [p.copy() for p in partitions(6, k=2)] == [{2: 3}, \
{1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=3)] == [{3: 2}, \
{1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2}, \
{1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=2, m=2)] == []
assert [p.copy() for p in partitions(8, k=4, m=3)] == [{4: 2},\
{1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}]
def test_partitions():
assert [p.copy() for p in partitions(6, k=2)] == [{2: 3}, \
{1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=3)] == [{3: 2}, \
{1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2}, \
{1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=2, m=2)] == []
assert [p.copy() for p in partitions(8, k=4, m=3)] == [{4: 2},\
{1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}]
def test_partitions():
assert [p.copy() for p in partitions(6, k=2)] == [{2: 3}, \
{1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=3)] == [{3: 2}, \
{1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2}, \
{1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=2, m=2)] == []
assert [p.copy() for p in partitions(8, k=4, m=3)] == [{4: 2},\
{1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}]
assert [p.copy() for p in partitions(S(3), 2)] == \
[{3: 1}, {1: 1, 2: 1}]
raises(ValueError, 'list(partitions(3, 0))')
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq("a")) == ["a"]
assert list(uniq("ababc")) == list("abc")
assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1]]
assert list(uniq(permutations(i for i in [[1], 2, 2]))) == [([1], 2, 2), (2, [1], 2), (2, 2, [1])]
assert list(uniq([2, 3, 2, 4, [2], [1], [2], [3], [1]])) == [2, 3, 4, [2], [1], [3]]
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == \
[{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq('a')) == ['a']
assert list(uniq('ababc')) == list('abc')
assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1], [1]]
assert list(uniq(permutations(i for i in [[1], 2, 2]))) == \
[([1], 2, 2), (2, [1], 2), (2, 2, [1]), (2, [1], 2), (2, 2, [1])]
def fourth_term(lambdas: tp.Dict[int, int]) -> tp.Any:
min_lambda = min(lambdas)
if min_lambda < 2:
return 0
reduced_lambdas = lambdas.copy()
decrease_key(reduced_lambdas, min_lambda)
return t / Q * sum(
map(lambda mu: subterm(mu, reduced_lambdas, f_1),
partitions(min_lambda - 2)))
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == \
[{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq('a')) == ['a']
assert list(uniq('ababc')) == list('abc')
assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1], [1]]
assert list(uniq(permutations(i for i in [[1], 2, 2]))) == \
[([1], 2, 2), (2, [1], 2), (2, 2, [1]), (2, [1], 2), (2, 2, [1])]
def first_term(lambdas: tp.Dict[int, int]) -> tp.Any:
min_lambda = min(lambdas)
if min_lambda == 1:
return 0
reduced_lambdas = lambdas.copy()
decrease_key(reduced_lambdas, min_lambda)
return -t / Q * q ** 2 * (1 - q) * \
sum(map(lambda mu: subterm(mu, reduced_lambdas, f_1),
partitions(min_lambda, k=min_lambda-1)))
def rs_partitions(N, n):
from sympy.utilities.iterables import multiset_permutations, partitions
k = 2 * n
for m, p in partitions(N, m=k, size=True):
if m >= n:
for q in multiset_permutations(
sum([[k] * v for k, v in p.items()], []) + [0] * (k - m)):
res = [(q[:n][i], q[n:][i]) for i in range(n)]
if not (0 in map(lambda x: x[0] or x[1], res)):
yield res
def test_nD_derangements():
from sympy.utilities.iterables import (partitions, multiset,
multiset_derangements,
multiset_permutations)
from sympy.functions.combinatorial.numbers import nD
got = []
for i in partitions(8, k=4):
s = []
it = 0
for k, v in i.items():
for i in range(v):
s.extend([it] * k)
it += 1
ms = multiset(s)
c1 = sum(1 for i in multiset_permutations(s)
if all(i != j for i, j in zip(i, s)))
assert c1 == nD(ms) == nD(ms, 0) == nD(ms, 1)
v = [tuple(i) for i in multiset_derangements(s)]
c2 = len(v)
assert c2 == len(set(v))
assert c1 == c2
got.append(c1)
assert got == [
1, 4, 6, 12, 24, 24, 61, 126, 315, 780, 297, 772, 2033, 5430, 14833
]
assert nD('1112233456', brute=True) == nD('1112233456') == 16356
assert nD('') == nD([]) == nD({}) == 0
assert nD({1: 0}) == 0
raises(ValueError, lambda: nD({1: -1}))
assert nD('112') == 0
assert nD(i='112') == 0
assert [nD(n=i) for i in range(6)] == [0, 0, 1, 2, 9, 44]
assert nD((i for i in range(4))) == nD('0123') == 9
assert nD(m=(i for i in range(4))) == 3
assert nD(m={0: 1, 1: 1, 2: 1, 3: 1}) == 3
assert nD(m=[0, 1, 2, 3]) == 3
raises(TypeError, lambda: nD(m=0))
raises(TypeError, lambda: nD(-1))
assert nD({-1: 1, -2: 1}) == 1
assert nD(m={0: 3}) == 0
raises(ValueError, lambda: nD(i='123', n=3))
raises(ValueError, lambda: nD(i='123', m=(1, 2)))
raises(ValueError, lambda: nD(n=0, m=(1, 2)))
raises(ValueError, lambda: nD({1: -1}))
raises(ValueError, lambda: nD(m={-1: 1, 2: 1}))
raises(ValueError, lambda: nD(m={1: -1, 2: 1}))
raises(ValueError, lambda: nD(m=[-1, 2]))
raises(TypeError, lambda: nD({1: x}))
raises(TypeError, lambda: nD(m={1: x}))
raises(TypeError, lambda: nD(m={x: 1}))
def fifth_term(lambdas: tp.Dict[int, int]) -> tp.Any:
min_lambda = min(lambdas)
reduced_lambdas = lambdas.copy()
decrease_key(reduced_lambdas, min_lambda)
answer = 0
for nu, sub_lambda in partition_into_two(reduced_lambdas):
if min_lambda + norm(nu) - 2 < 0:
continue
combinations = 1
for key in nu:
combinations *= sm.binomial(reduced_lambdas[key], nu[key])
answer += combinations * map_prod(nu, f_3) * \
sum(map(lambda mu: subterm(mu, sub_lambda, f_2),
partitions(min_lambda + norm(nu) - 2)))
return q * Q * answer
def test_uniq():
assert list(uniq(p for p in partitions(4))) == \
[{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq('a')) == ['a']
assert list(uniq('ababc')) == list('abc')
assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1]]
assert list(uniq(permutations(i for i in [[1], 2, 2]))) == \
[([1], 2, 2), (2, [1], 2), (2, 2, [1])]
assert list(uniq([2, 3, 2, 4, [2], [1], [2], [3], [1]])) == \
[2, 3, 4, [2], [1], [3]]
f = [1]
raises(RuntimeError, lambda: [f.remove(i) for i in uniq(f)])
f = [[1]]
raises(RuntimeError, lambda: [f.remove(i) for i in uniq(f)])
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == [
{4: 1},
{1: 1, 3: 1},
{2: 2},
{1: 2, 2: 1},
{1: 4},
]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq("a")) == ["a"]
assert list(uniq("ababc")) == list("abc")
assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1]]
assert list(uniq(permutations(i for i in [[1], 2, 2]))) == [
([1], 2, 2),
(2, [1], 2),
(2, 2, [1]),
]
assert list(uniq([2, 3, 2, 4, [2], [1], [2], [3], [1]])) == [2, 3, 4, [2], [1], [3]]
def test_integer_partition():
# no zeros in partition
raises(ValueError, lambda: IntegerPartition(list(range(3))))
# check fails since 1 + 2 != 100
raises(ValueError, lambda: IntegerPartition(100, list(range(1, 3))))
a = IntegerPartition(8, [1, 3, 4])
b = a.next_lex()
c = IntegerPartition([1, 3, 4])
d = IntegerPartition(8, {1: 3, 3: 1, 2: 1})
assert a == c
assert a.integer == d.integer
assert a.conjugate == [3, 2, 2, 1]
assert (a == b) is False
assert a <= b
assert (a > b) is False
assert a != b
for i in range(1, 11):
next = set()
prev = set()
a = IntegerPartition([i])
ans = set([IntegerPartition(p) for p in partitions(i)])
n = len(ans)
for j in range(n):
next.add(a)
a = a.next_lex()
IntegerPartition(i, a.partition) # check it by giving i
for j in range(n):
prev.add(a)
a = a.prev_lex()
IntegerPartition(i, a.partition) # check it by giving i
assert next == ans
assert prev == ans
assert IntegerPartition([1, 2, 3]).as_ferrers() == '###\n##\n#'
assert IntegerPartition([1, 1, 3]).as_ferrers('o') == 'ooo\no\no'
assert str(IntegerPartition([1, 1, 3])) == '[3, 1, 1]'
assert IntegerPartition([1, 1, 3]).partition == [3, 1, 1]
raises(ValueError, lambda: random_integer_partition(-1))
assert random_integer_partition(1) == [1]
assert random_integer_partition(10, seed=[1, 3, 2, 1, 5, 1]
) == [5, 2, 1, 1, 1]
def test_integer_partition():
# no zeros in partition
raises(ValueError, lambda: IntegerPartition(list(range(3))))
# check fails since 1 + 2 != 100
raises(ValueError, lambda: IntegerPartition(100, list(range(1, 3))))
a = IntegerPartition(8, [1, 3, 4])
b = a.next_lex()
c = IntegerPartition([1, 3, 4])
d = IntegerPartition(8, {1: 3, 3: 1, 2: 1})
assert a == c
assert a.integer == d.integer
assert a.conjugate == [3, 2, 2, 1]
assert (a == b) is False
assert a <= b
assert (a > b) is False
assert a != b
for i in range(1, 11):
next = set()
prev = set()
a = IntegerPartition([i])
ans = set([IntegerPartition(p) for p in partitions(i)])
n = len(ans)
for j in range(n):
next.add(a)
a = a.next_lex()
IntegerPartition(i, a.partition) # check it by giving i
for j in range(n):
prev.add(a)
a = a.prev_lex()
IntegerPartition(i, a.partition) # check it by giving i
assert next == ans
assert prev == ans
assert IntegerPartition([1, 2, 3]).as_ferrers() == '###\n##\n#'
assert IntegerPartition([1, 1, 3]).as_ferrers('o') == 'ooo\no\no'
assert str(IntegerPartition([1, 1, 3])) == '[3, 1, 1]'
assert IntegerPartition([1, 1, 3]).partition == [3, 1, 1]
raises(ValueError, lambda: random_integer_partition(-1))
assert random_integer_partition(1) == [1]
assert random_integer_partition(10, seed=[1, 3, 2, 1, 5,
1]) == [5, 2, 1, 1, 1]
def calcFaaDiBrunoFromPartitionsFromFunction( baseDerivativeFunction, l, n, x, thresholdForExp=30.0, verbose=True):
"""
Function to calculate derivatives of iterated base function
:param baseDerivativeFunction: Function that returns derivaties of base function
:param l: Order of function iteration
:param n: Highest order of derivative to be evaluated for iterated function
:param x: Point at which derivatives are to be evaluated
:param thresholdForExp: Threshold beyond which differences in logarithms will be capped
:param verbose: Boolean flag. If true then print messages
:return: Dictionary containing logarithms and signs of derivates of iterated base function
"""
# set up arrays for holding derivatives (on log scale)
# of the base function, the current iteration and next iteration
evaluationPoint = x
baseDerivativesLog = np.zeros( n )
baseDerivativesSign = np.ones( n )
currentDerivativesLog = np.zeros( n )
currentDerivativesSign = np.ones( n )
for k in range( n ):
baseDerivativeTmp = baseDerivativeFunction( x, k+1 )
if( np.abs( baseDerivativeTmp ) > 1.0e-12 ):
baseDerivativesLog[k] = np.log( np.abs(baseDerivativeTmp) )
baseDerivativesSign[k] = np.sign( baseDerivativeTmp)
# get current derivatives
currentDerivativesLog[k] = np.log( np.abs( baseDerivativeTmp ) )
currentDerivativesSign[k] = np.sign( baseDerivativeTmp )
else:
baseDerivativesLog[k] = float( '-Inf' )
baseDerivativesSign[k] = 1.0
currentDerivativesLog[k] = float( '-Inf' )
currentDerivativesSign[k] = 1.0
nextDerivativesLog = np.zeros( n )
nextDerivativesSign = np.ones( n )
# initialize array for holding derivatives at each iteration
iteratedFunctionDerivativesLog = np.zeros( (l, n) )
iteratedFunctionDerivativesSign = np.ones( (l, n) )
# store base derivatives in first row of array
iteratedFunctionDerivativesLog[0, :] = baseDerivativesLog
iteratedFunctionDerivativesSign[0, :] = baseDerivativesSign
# set number of function iterations
nIterations = l-1
# if we need to iterate then do so
if( nIterations > 0 ):
# evaluate approximate number of paritions required
log_nPartitions = calcHardyRamanujanApprox( n )
if( verbose==True ):
print( "You have requested evaluation up to derivative " + str(n) )
if( log_nPartitions > np.log( 1.0e6 ) ):
print( "Warning: There are approximately " + str(int(np.round(np.exp(log_nPartitions)))) + " partitions of " + str(n) )
print( "Warning: Evaluation will be costly both in terms of memory and run-time" )
# store partitions
pStore = {}
for k in range( n ):
# get partition iterator
pIterator = partitions(k+1)
pStore[k] = [p.copy() for p in pIterator]
# loop over function iterations
for iteration in range( nIterations ):
evaluationPoint = baseDerivativeFunction( evaluationPoint, 0 )
if( verbose==True ):
print( "Evaluating derivatives for function iteration " + str(iteration+1) )
for k in range( n ):
faaSumLog = float( '-Inf' )
faaSumSign = 1
# get partitions
partitionsK = pStore[k]
for pidx in range( len(partitionsK) ):
p = partitionsK[pidx]
sumTmp = 0.0
sumMultiplicty = 0
parityTmp = 1
for i in p.keys():
value = float(i)
multiplicity = float( p[i] )
sumMultiplicty += p[i]
sumTmp += multiplicity * currentDerivativesLog[i-1]
sumTmp -= gammaln( multiplicity + 1.0 )
sumTmp -= multiplicity * gammaln( value + 1.0 )
parityTmp *= np.power( currentDerivativesSign[i-1], multiplicity )
#evaluationPointTmp = np.exp( currentDerivativesLog[0] ) * currentDerivativesSign[0]
baseDerivativeTmp = baseDerivativeFunction( evaluationPoint , sumMultiplicty )
if( np.abs( baseDerivativeTmp ) > 1.0e-12 ):
sumTmp += np.log( np.abs( baseDerivativeTmp ) )
parityTmp *= np.sign( baseDerivativeTmp )
else:
sumTmp = float( '-Inf' )
parityTmp = 1.0
# now update faaSum on log scale
if( sumTmp > float( '-Inf' ) ):
if( faaSumLog > float( '-Inf' ) ):
diffLog = sumTmp - faaSumLog
if( np.abs(diffLog) <= thresholdForExp ):
if( diffLog >= 0.0 ):
faaSumLog = sumTmp
faaSumLog += np.log( 1.0 + (float(parityTmp*faaSumSign) * np.exp( -diffLog )) )
faaSumSign = parityTmp
else:
faaSumLog += np.log( 1.0 + (float(parityTmp*faaSumSign) * np.exp( diffLog )) )
else:
if( diffLog > thresholdForExp ):
faaSumLog = sumTmp
faaSumSign = parityTmp
else:
faaSumLog = sumTmp
faaSumSign = parityTmp
nextDerivativesLog[k] = faaSumLog + gammaln( float(k+2) )
nextDerivativesSign[k] = faaSumSign
# update accounting for proceeding to next function iteration
currentDerivativesLog[0:n] = nextDerivativesLog[0:n]
currentDerivativesSign[0:n] = nextDerivativesSign[0:n]
iteratedFunctionDerivativesLog[iteration+1, 0:n] = currentDerivativesLog[0:n]
iteratedFunctionDerivativesSign[iteration+1, 0:n] = currentDerivativesSign[0:n]
return( {'logDerivatives':iteratedFunctionDerivativesLog, 'signDerivatives':iteratedFunctionDerivativesSign} )
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == \
[{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq('a')) == ['a']
assert list(uniq('ababc')) == list('abc')
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_partitions():
ans = [[{}], [(0, {})]]
for i in range(2):
assert list(partitions(0, size=i)) == ans[i]
assert list(partitions(1, 0, size=i)) == ans[i]
assert list(partitions(6, 2, 2, size=i)) == ans[i]
assert list(partitions(6, 2, None, size=i)) != ans[i]
assert list(partitions(6, None, 2, size=i)) != ans[i]
assert list(partitions(6, 2, 0, size=i)) == ans[i]
assert [p for p in partitions(6, k=2)] == [{
2: 3
}, {
1: 2,
2: 2
}, {
1: 4,
2: 1
}, {
1: 6
}]
assert [p for p in partitions(6, k=3)] == [{
3: 2
}, {
1: 1,
2: 1,
3: 1
}, {
1: 3,
3: 1
}, {
2: 3
}, {
1: 2,
2: 2
}, {
1: 4,
2: 1
}, {
1: 6
}]
assert [p for p in partitions(8, k=4, m=3)] == [{
4: 2
}, {
1: 1,
3: 1,
4: 1
}, {
2: 2,
4: 1
}, {
2: 1,
3: 2
}] == [
i for i in partitions(8, k=4, m=3)
if all(k <= 4 for k in i) and sum(i.values()) <= 3
]
assert [p for p in partitions(S(3), m=2)] == [{3: 1}, {1: 1, 2: 1}]
assert [i for i in partitions(4, k=3)] == [{
1: 1,
3: 1
}, {
2: 2
}, {
1: 2,
2: 1
}, {
1: 4
}] == [i for i in partitions(4) if all(k <= 3 for k in i)]
# Consistency check on output of _partitions and RGS_unrank.
# This provides a sanity test on both routines. Also verifies that
# the total number of partitions is the same in each case.
# (from pkrathmann2)
for n in range(2, 6):
i = 0
for m, q in _set_partitions(n):
assert q == RGS_unrank(i, n)
i += 1
assert i == RGS_enum(n)
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq("a")) == ["a"]
assert list(uniq("ababc")) == list("abc")
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == \
[{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq('a')) == ['a']
assert list(uniq('ababc')) == list('abc')
def test_partitions():
ans = [[{}], [(0, {})]]
for i in range(2):
assert list(partitions(0, size=i)) == ans[i]
assert list(partitions(1, 0, size=i)) == ans[i]
assert list(partitions(6, 2, 2, size=i)) == ans[i]
assert list(partitions(6, 2, None, size=i)) != ans[i]
assert list(partitions(6, None, 2, size=i)) != ans[i]
assert list(partitions(6, 2, 0, size=i)) == ans[i]
assert [p.copy() for p in partitions(6, k=2)] == [
{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=3)] == [
{3: 2}, {1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2},
{1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(8, k=4, m=3)] == [
{4: 2}, {1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}] == [
i.copy() for i in partitions(8, k=4, m=3) if all(k <= 4 for k in i)
and sum(i.values()) <=3]
assert [p.copy() for p in partitions(S(3), m=2)] == [
{3: 1}, {1: 1, 2: 1}]
assert [i.copy() for i in partitions(4, k=3)] == [
{1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}] == [
i.copy() for i in partitions(4) if all(k <= 3 for k in i)]
# Consistency check on output of _partitions and RGS_unrank.
# This provides a sanity test on both routines. Also verifies that
# the total number of partitions is the same in each case.
# (from pkrathmann2)
for n in range(2, 6):
i = 0
for m, q in _set_partitions(n):
assert q == RGS_unrank(i, n)
i += 1
assert i == RGS_enum(n)
def test_F6():
partTest = [p.copy() for p in partitions(4)]
partDesired = [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert partTest == partDesired
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'}))
def test_F6():
partTest = [p.copy() for p in partitions(4)]
partDesired = [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2:1}, {1: 4}]
assert partTest == partDesired
def kbin(l, k, ordered=True):
"""
Return sequence ``l`` partitioned into ``k`` bins.
If ordered is True then the order of the items in the
flattened partition will be the same as the order of the
items in ``l``; if False, all permutations of the items will
be given; if None, only unique permutations for a given
partition will be given.
Examples
========
>>> from sympy.utilities.iterables import kbin
>>> for p in kbin(range(3), 2):
... print p
...
[[0], [1, 2]]
[[0, 1], [2]]
>>> for p in kbin(range(3), 2, ordered=False):
... print p
...
[(0,), (1, 2)]
[(0,), (2, 1)]
[(1,), (0, 2)]
[(1,), (2, 0)]
[(2,), (0, 1)]
[(2,), (1, 0)]
[(0, 1), (2,)]
[(0, 2), (1,)]
[(1, 0), (2,)]
[(1, 2), (0,)]
[(2, 0), (1,)]
[(2, 1), (0,)]
>>> for p in kbin(range(3), 2, ordered=None):
... print p
...
[[0], [1, 2]]
[[1], [2, 0]]
[[2], [0, 1]]
[[0, 1], [2]]
[[1, 2], [0]]
[[2, 0], [1]]
"""
from sympy.utilities.iterables import partitions
from itertools import permutations
def rotations(seq):
for i in range(len(seq)):
yield seq
seq.append(seq.pop(0))
if ordered is None:
func = rotations
else:
func = permutations
for p in partitions(len(l), k):
if sum(p.values()) != k:
continue
for pe in permutations(p.keys()):
rv = []
i = 0
for part in pe:
for do in range(p[part]):
j = i + part
rv.append(l[i: j])
i = j
if ordered:
yield rv
else:
template = [len(i) for i in rv]
for pp in func(l):
rvp = []
ii = 0
for t in template:
jj = ii + t
rvp.append(pp[ii: jj])
ii = jj
yield rvp
