Returns the cartesian product of sequences as a generator.
>>> from sympy.utilities.iterables import cartes
>>> list(cartes([1,2,3], 'ab'))
[(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b'), (3, 'a'), (3, 'b')]
variations(seq, n) Returns all the variations of the list of size n.
Has an optional third argument. Must be a boolean value and makes the method return the variations with repetition if set to True, or the variations without repetition if set to False.
>>> from sympy.utilities.iterables import variations
>>> list(variations([1,2,3], 2))
[(1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2)]
>>> list(variations([1,2,3], 2, True))
[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)]
Although the combinatorics module contains Partition and IntegerPartition classes for investigation and manipulation of partitions, there are a few functions to generate partitions that can be used as lowlevel tools for routines: partitions and multiset_partitions. The former gives integer partitions, and the latter gives enumerated partitions of elements. There is also a routine kbins that will give a variety of permutations of partions.
partitions:
>>> from sympy.utilities.iterables import partitions
>>> [p.copy() for s, p in partitions(7, m=2, size=True) if s == 2]
[{1: 1, 6: 1}, {2: 1, 5: 1}, {3: 1, 4: 1}]
multiset_partitions:
>>> from sympy.utilities.iterables import multiset_partitions
>>> [p for p in multiset_partitions(3, 2)]
[[[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]]]
>>> [p for p in multiset_partitions([1, 1, 1, 2], 2)]
[[[1, 1, 1], [2]], [[1, 1, 2], [1]], [[1, 1], [1, 2]]]
kbins:
>>> from sympy.utilities.iterables import kbins
>>> def show(k):
... rv = []
... for p in k:
... rv.append(','.join([''.join(j) for j in p]))
... return sorted(rv)
...
>>> show(kbins("ABCD", 2))
['A,BCD', 'AB,CD', 'ABC,D']
>>> show(kbins("ABC", 2))
['A,BC', 'AB,C']
>>> show(kbins("ABC", 2, ordered=0)) # same as multiset_partitions
['A,BC', 'AB,C', 'AC,B']
>>> show(kbins("ABC", 2, ordered=1))
['A,BC', 'A,CB',
'B,AC', 'B,CA',
'C,AB', 'C,BA']
>>> show(kbins("ABC", 2, ordered=10))
['A,BC', 'AB,C', 'AC,B',
'B,AC', 'BC,A',
'C,AB']
>>> show(kbins("ABC", 2, ordered=11))
['A,BC', 'A,CB', 'AB,C', 'AC,B',
'B,AC', 'B,CA', 'BA,C', 'BC,A',
'C,AB', 'C,BA', 'CA,B', 'CB,A']
Generates the binary partition of n.
A binary partition consists only of numbers that are powers of two. Each step reduces a 2**(k+1) to 2**k and 2**k. Thus 16 is converted to 8 and 8.
Reference: TAOCP 4, section 7.2.1.5, problem 64
Examples
>>> from sympy.utilities.iterables import binary_partitions
>>> for i in binary_partitions(5):
... print(i)
...
[4, 1]
[2, 2, 1]
[2, 1, 1, 1]
[1, 1, 1, 1, 1]
Wrapper to necklaces to return a free (unrestricted) necklace.
Return the printed output of func().
\(func\) should be a function without arguments that produces output with print statements.
>>> from sympy.utilities.iterables import capture
>>> from sympy import pprint
>>> from sympy.abc import x
>>> def foo():
... print('hello world!')
...
>>> 'hello' in capture(foo) # foo, not foo()
True
>>> capture(lambda: pprint(2/x))
'2\n\nx\n'
Return the subsequence that is a common start of sequences in seqs.
>>> from sympy.utilities.iterables import common_prefix
>>> common_prefix(list(range(3)))
[0, 1, 2]
>>> common_prefix(list(range(3)), list(range(4)))
[0, 1, 2]
>>> common_prefix([1, 2, 3], [1, 2, 5])
[1, 2]
>>> common_prefix([1, 2, 3], [1, 3, 5])
[1]
Return the subsequence that is a common ending of sequences in seqs.
>>> from sympy.utilities.iterables import common_suffix
>>> common_suffix(list(range(3)))
[0, 1, 2]
>>> common_suffix(list(range(3)), list(range(4)))
[]
>>> common_suffix([1, 2, 3], [9, 2, 3])
[2, 3]
>>> common_suffix([1, 2, 3], [9, 7, 3])
[3]
Only yield elements from \(iterator\) that do not occur in \(exclude\).
Parameters :  iterator : iterable iterator to take elements from : exclude : iterable elements to exclude : 

Returns :  iterator : iterator filtered iterator : 
Recursively denest iterable containers.
>>> from sympy.utilities.iterables import flatten
>>> flatten([1, 2, 3])
[1, 2, 3]
>>> flatten([1, 2, [3]])
[1, 2, 3]
>>> flatten([1, [2, 3], [4, 5]])
[1, 2, 3, 4, 5]
>>> flatten([1.0, 2, (1, None)])
[1.0, 2, 1, None]
If you want to denest only a specified number of levels of nested containers, then set levels flag to the desired number of levels:
>>> ls = [[(2, 1), (1, 2)], [(0, 0)]]
>>> flatten(ls, levels=1)
[(2, 1), (1, 2), (0, 0)]
If cls argument is specified, it will only flatten instances of that class, for example:
>>> from sympy.core import Basic
>>> class MyOp(Basic):
... pass
...
>>> flatten([MyOp(1, MyOp(2, 3))], cls=MyOp)
[1, 2, 3]
adapted from http://kogswww.informatik.unihamburg.de/~meine/python_tricks
Return permutations of [0, 1, ..., n  1] such that each permutation differs from the last by the exchange of a single pair of neighbors. The n! permutations are returned as an iterator. In order to obtain the next permutation from a random starting permutation, use the next_trotterjohnson method of the Permutation class (which generates the same sequence in a different manner).
See also
sympy.combinatorics.Permutation.next_trotterjohnson
References
Examples
>>> from itertools import permutations
>>> from sympy.utilities.iterables import generate_bell
>>> from sympy import zeros, Matrix
This is the sort of permutation used in the ringing of physical bells, and does not produce permutations in lexicographical order. Rather, the permutations differ from each other by exactly one inversion, and the position at which the swapping occurs varies periodically in a simple fashion. Consider the first few permutations of 4 elements generated by permutations and generate_bell:
>>> list(permutations(range(4)))[:5]
[(0, 1, 2, 3), (0, 1, 3, 2), (0, 2, 1, 3), (0, 2, 3, 1), (0, 3, 1, 2)]
>>> list(generate_bell(4))[:5]
[(0, 1, 2, 3), (0, 1, 3, 2), (0, 3, 1, 2), (3, 0, 1, 2), (3, 0, 2, 1)]
Notice how the 2nd and 3rd lexicographical permutations have 3 elements out of place whereas each “bell” permutation always has only two elements out of place relative to the previous permutation (and so the signature (+/1) of a permutation is opposite of the signature of the previous permutation).
How the position of inversion varies across the elements can be seen by tracing out where the largest number appears in the permutations:
>>> m = zeros(4, 24)
>>> for i, p in enumerate(generate_bell(4)):
... m[:, i] = Matrix([j  3 for j in list(p)]) # make largest zero
>>> m.print_nonzero('X')
[XXX XXXXXX XXXXXX XXX]
[XX XX XXXX XX XXXX XX XX]
[X XXXX XX XXXX XX XXXX X]
[ XXXXXX XXXXXX XXXXXX ]
Routine to generate unique derangements.
TODO: This will be rewritten to use the ECO operator approach once the permutations branch is in master.
See also
sympy.functions.combinatorial.factorials.subfactorial
Examples
>>> from sympy.utilities.iterables import generate_derangements
>>> list(generate_derangements([0, 1, 2]))
[[1, 2, 0], [2, 0, 1]]
>>> list(generate_derangements([0, 1, 2, 3]))
[[1, 0, 3, 2], [1, 2, 3, 0], [1, 3, 0, 2], [2, 0, 3, 1], [2, 3, 0, 1], [2, 3, 1, 0], [3, 0, 1, 2], [3, 2, 0, 1], [3, 2, 1, 0]]
>>> list(generate_derangements([0, 1, 1]))
[]
Generates involutions.
An involution is a permutation that when multiplied by itself equals the identity permutation. In this implementation the involutions are generated using Fixed Points.
Alternatively, an involution can be considered as a permutation that does not contain any cycles with a length that is greater than two.
Reference: http://mathworld.wolfram.com/PermutationInvolution.html
Examples
>>> from sympy.utilities.iterables import generate_involutions
>>> list(generate_involutions(3))
[(0, 1, 2), (0, 2, 1), (1, 0, 2), (2, 1, 0)]
>>> len(list(generate_involutions(4)))
10
This algorithm generates oriented forests.
An oriented graph is a directed graph having no symmetric pair of directed edges. A forest is an acyclic graph, i.e., it has no cycles. A forest can also be described as a disjoint union of trees, which are graphs in which any two vertices are connected by exactly one simple path.
Reference: [1] T. Beyer and S.M. Hedetniemi: constant time generation of rooted trees, SIAM J. Computing Vol. 9, No. 4, November 1980 [2] http://stackoverflow.com/questions/1633833/orientedforesttaocpalgorithminpython
Examples
>>> from sympy.utilities.iterables import generate_oriented_forest
>>> list(generate_oriented_forest(4))
[[0, 1, 2, 3], [0, 1, 2, 2], [0, 1, 2, 1], [0, 1, 2, 0], [0, 1, 1, 1], [0, 1, 1, 0], [0, 1, 0, 1], [0, 1, 0, 0], [0, 0, 0, 0]]
Splits a sequence into a list of lists of equal, adjacent elements.
See also
Examples
>>> from sympy.utilities.iterables import group
>>> group([1, 1, 1, 2, 2, 3])
[[1, 1, 1], [2, 2], [3]]
>>> group([1, 1, 1, 2, 2, 3], multiple=False)
[(1, 3), (2, 2), (3, 1)]
>>> group([1, 1, 3, 2, 2, 1], multiple=False)
[(1, 2), (3, 1), (2, 2), (1, 1)]
Return True if there are any duplicate elements in seq.
Examples
>>> from sympy.utilities.iterables import has_dups
>>> from sympy import Dict, Set
>>> has_dups((1, 2, 1))
True
>>> has_dups(range(3))
False
>>> all(has_dups(c) is False for c in (set(), Set(), dict(), Dict()))
True
Return True if there are any different elements in seq.
Examples
>>> from sympy.utilities.iterables import has_variety
>>> has_variety((1, 2, 1))
True
>>> has_variety((1, 1, 1))
False
Return a list of length bits corresponding to the binary value of n with small bits to the right (last). If bits is omitted, the length will be the number required to represent n. If the bits are desired in reversed order, use the [::1] slice of the returned list.
If a sequence of all bitslength lists starting from [0, 0,..., 0] through [1, 1, ..., 1] are desired, pass a noninteger for bits, e.g. ‘all’.
If the bit string is desired pass str=True.
Examples
>>> from sympy.utilities.iterables import ibin
>>> ibin(2)
[1, 0]
>>> ibin(2, 4)
[0, 0, 1, 0]
>>> ibin(2, 4)[::1]
[0, 1, 0, 0]
If all lists corresponding to 0 to 2**n  1, pass a noninteger for bits:
>>> bits = 2
>>> for i in ibin(2, 'all'):
... print(i)
(0, 0)
(0, 1)
(1, 0)
(1, 1)
If a bit string is desired of a given length, use str=True:
>>> n = 123
>>> bits = 10
>>> ibin(n, bits, str=True)
'0001111011'
>>> ibin(n, bits, str=True)[::1] # small bits left
'1101111000'
>>> list(ibin(3, 'all', str=True))
['000', '001', '010', '011', '100', '101', '110', '111']
Traverse a tree asking a user which branch to choose.
Return sequence l partitioned into k bins.
See also
Examples
>>> from sympy.utilities.iterables import kbins
The default is to give the items in the same order, but grouped into k partitions without any reordering:
>>> from __future__ import print_function
>>> for p in kbins(list(range(5)), 2):
... print(p)
...
[[0], [1, 2, 3, 4]]
[[0, 1], [2, 3, 4]]
[[0, 1, 2], [3, 4]]
[[0, 1, 2, 3], [4]]
The ordered flag which is either None (to give the simple partition of the the elements) or is a 2 digit integer indicating whether the order of the bins and the order of the items in the bins matters. Given:
A = [[0], [1, 2]]
B = [[1, 2], [0]]
C = [[2, 1], [0]]
D = [[0], [2, 1]]
the following values for ordered have the shown meanings:
00 means A == B == C == D
01 means A == B
10 means A == D
11 means A == A
>>> for ordered in [None, 0, 1, 10, 11]:
... print('ordered = %s' % ordered)
... for p in kbins(list(range(3)), 2, ordered=ordered):
... print(' %s' % p)
...
ordered = None
[[0], [1, 2]]
[[0, 1], [2]]
ordered = 0
[[0, 1], [2]]
[[0, 2], [1]]
[[0], [1, 2]]
ordered = 1
[[0], [1, 2]]
[[0], [2, 1]]
[[1], [0, 2]]
[[1], [2, 0]]
[[2], [0, 1]]
[[2], [1, 0]]
ordered = 10
[[0, 1], [2]]
[[2], [0, 1]]
[[0, 2], [1]]
[[1], [0, 2]]
[[0], [1, 2]]
[[1, 2], [0]]
ordered = 11
[[0], [1, 2]]
[[0, 1], [2]]
[[0], [2, 1]]
[[0, 2], [1]]
[[1], [0, 2]]
[[1, 0], [2]]
[[1], [2, 0]]
[[1, 2], [0]]
[[2], [0, 1]]
[[2, 0], [1]]
[[2], [1, 0]]
[[2, 1], [0]]
Return a tuple where the smallest element appears first; if directed is True (default) then the order is preserved, otherwise the sequence will be reversed if that gives a smaller ordering.
If every element appears only once then is_set can be set to True for more efficient processing.
If the smallest element is known at the time of calling, it can be passed and the calculation of the smallest element will be omitted.
Examples
>>> from sympy.combinatorics.polyhedron import minlex
>>> minlex((1, 2, 0))
(0, 1, 2)
>>> minlex((1, 0, 2))
(0, 2, 1)
>>> minlex((1, 0, 2), directed=False)
(0, 1, 2)
>>> minlex('11010011000', directed=True)
'00011010011'
>>> minlex('11010011000', directed=False)
'00011001011'
Return the hashable sequence in multiset form with values being the multiplicity of the item in the sequence.
See also
Examples
>>> from sympy.utilities.iterables import multiset
>>> multiset('mississippi')
{'i': 4, 'm': 1, 'p': 2, 's': 4}
Return the unique combinations of size n from multiset m.
Examples
>>> from sympy.utilities.iterables import multiset_combinations
>>> from itertools import combinations
>>> [''.join(i) for i in multiset_combinations('baby', 3)]
['abb', 'aby', 'bby']
>>> def count(f, s): return len(list(f(s, 3)))
The number of combinations depends on the number of letters; the number of unique combinations depends on how the letters are repeated.
>>> s1 = 'abracadabra'
>>> s2 = 'banana tree'
>>> count(combinations, s1), count(multiset_combinations, s1)
(165, 23)
>>> count(combinations, s2), count(multiset_combinations, s2)
(165, 54)
Return unique partitions of the given multiset (in list form). If m is None, all multisets will be returned, otherwise only partitions with m parts will be returned.
If multiset is an integer, a range [0, 1, ..., multiset  1] will be supplied.
See also
partitions, sympy.combinatorics.partitions.Partition, sympy.combinatorics.partitions.IntegerPartition, sympy.functions.combinatorial.numbers.nT
Notes
When all the elements are the same in the multiset, the order of the returned partitions is determined by the partitions routine. If one is counting partitions then it is better to use the nT function.
Examples
>>> from sympy.utilities.iterables import multiset_partitions
>>> list(multiset_partitions([1, 2, 3, 4], 2))
[[[1, 2, 3], [4]], [[1, 2, 4], [3]], [[1, 2], [3, 4]],
[[1, 3, 4], [2]], [[1, 3], [2, 4]], [[1, 4], [2, 3]],
[[1], [2, 3, 4]]]
>>> list(multiset_partitions([1, 2, 3, 4], 1))
[[[1, 2, 3, 4]]]
Only unique partitions are returned and these will be returned in a canonical order regardless of the order of the input:
>>> a = [1, 2, 2, 1]
>>> ans = list(multiset_partitions(a, 2))
>>> a.sort()
>>> list(multiset_partitions(a, 2)) == ans
True
>>> a = range(3, 1, 1)
>>> (list(multiset_partitions(a)) ==
... list(multiset_partitions(sorted(a))))
True
If m is omitted then all partitions will be returned:
>>> list(multiset_partitions([1, 1, 2]))
[[[1, 1, 2]], [[1, 1], [2]], [[1, 2], [1]], [[1], [1], [2]]]
>>> list(multiset_partitions([1]*3))
[[[1, 1, 1]], [[1], [1, 1]], [[1], [1], [1]]]
Counting
The number of partitions of a set is given by the bell number:
>>> from sympy import bell
>>> len(list(multiset_partitions(5))) == bell(5) == 52
True
The number of partitions of length k from a set of size n is given by the Stirling Number of the 2nd kind:
>>> def S2(n, k):
... from sympy import Dummy, binomial, factorial, Sum
... if k > n:
... return 0
... j = Dummy()
... arg = (1)**(kj)*j**n*binomial(k,j)
... return 1/factorial(k)*Sum(arg,(j,0,k)).doit()
...
>>> S2(5, 2) == len(list(multiset_partitions(5, 2))) == 15
True
These comments on counting apply to sets, not multisets.
Return the unique permutations of multiset m.
Examples
>>> from sympy.utilities.iterables import multiset_permutations
>>> from sympy import factorial
>>> [''.join(i) for i in multiset_permutations('aab')]
['aab', 'aba', 'baa']
>>> factorial(len('banana'))
720
>>> len(list(multiset_permutations('banana')))
60
A routine to generate necklaces that may (free=True) or may not (free=False) be turned over to be viewed. The “necklaces” returned are comprised of n integers (beads) with k different values (colors). Only unique necklaces are returned.
References
http://mathworld.wolfram.com/Necklace.html
Examples
>>> from sympy.utilities.iterables import necklaces, bracelets
>>> def show(s, i):
... return ''.join(s[j] for j in i)
The “unrestricted necklace” is sometimes also referred to as a “bracelet” (an object that can be turned over, a sequence that can be reversed) and the term “necklace” is used to imply a sequence that cannot be reversed. So ACB == ABC for a bracelet (rotate and reverse) while the two are different for a necklace since rotation alone cannot make the two sequences the same.
(mnemonic: Bracelets can be viewed Backwards, but Not Necklaces.)
>>> B = [show('ABC', i) for i in bracelets(3, 3)]
>>> N = [show('ABC', i) for i in necklaces(3, 3)]
>>> set(N)  set(B)
set(['ACB'])
>>> list(necklaces(4, 2))
[(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 1),
(0, 1, 0, 1), (0, 1, 1, 1), (1, 1, 1, 1)]
>>> [show('.o', i) for i in bracelets(4, 2)]
['....', '...o', '..oo', '.o.o', '.ooo', 'oooo']
Generate an infinite stream of Symbols consisting of a prefix and increasing subscripts provided that they do not occur in \(exclude\).
Parameters :  prefix : str, optional
cls : class, optional
start : int, optional


Returns :  sym : Symbol

Generate all partitions of integer n (>= 0).
Parameters :  ``m`` : integer (default gives partitions of all sizes)
``k`` : integer (default gives partitions number from 1 through n)
``size`` : bool (default False, only partition is returned)
Each partition is represented as a dictionary, mapping an integer : to the number of copies of that integer in the partition. For example, : the first partition of 4 returned is {4: 1}, “4: one of them”. : 

Examples
>>> from sympy.utilities.iterables import partitions
The numbers appearing in the partition (the key of the returned dict) are limited with k:
>>> for p in partitions(6, k=2):
... print(p)
{2: 3}
{1: 2, 2: 2}
{1: 4, 2: 1}
{1: 6}
The maximum number of parts in the partion (the sum of the values in the returned dict) are limited with m:
>>> for p in partitions(6, m=2):
... print(p)
...
{6: 1}
{1: 1, 5: 1}
{2: 1, 4: 1}
{3: 2}
Note that the _same_ dictionary object is returned each time. This is for speed: generating each partition goes quickly, taking constant time, independent of n.
>>> [p for p in partitions(6, k=2)]
[{1: 6}, {1: 6}, {1: 6}, {1: 6}]
If you want to build a list of the returned dictionaries then make a copy of them:
>>> [p.copy() for p in partitions(6, k=2)]
[{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
>>> [(M, p.copy()) for M, p in partitions(6, k=2, size=True)]
[(3, {2: 3}), (4, {1: 2, 2: 2}), (5, {1: 4, 2: 1}), (6, {1: 6})]
Generate all postfixes of a sequence.
Examples
>>> from sympy.utilities.iterables import postfixes
>>> list(postfixes([1,2,3,4]))
[[4], [3, 4], [2, 3, 4], [1, 2, 3, 4]]
Do a postorder traversal of a tree.
This generator recursively yields nodes that it has visited in a postorder fashion. That is, it descends through the tree depthfirst to yield all of a node’s children’s postorder traversal before yielding the node itself.
Parameters :  node : sympy expression
keys : (default None) sort key(s)


Examples
>>> from sympy.utilities.iterables import postorder_traversal
>>> from sympy.abc import w, x, y, z
The nodes are returned in the order that they are encountered unless key is given; simply passing key=True will guarantee that the traversal is unique.
>>> list(postorder_traversal(w + (x + y)*z))
[z, y, x, x + y, z*(x + y), w, w + z*(x + y)]
>>> list(postorder_traversal(w + (x + y)*z, keys=True))
[w, z, x, y, x + y, z*(x + y), w + z*(x + y)]
Yields
Generate all prefixes of a sequence.
Examples
>>> from sympy.utilities.iterables import prefixes
>>> list(prefixes([1,2,3,4]))
[[1], [1, 2], [1, 2, 3], [1, 2, 3, 4]]
Reshape the sequence according to the template in how.
Examples
>>> from sympy.utilities import reshape
>>> seq = list(range(1, 9))
>>> reshape(seq, [4]) # lists of 4
[[1, 2, 3, 4], [5, 6, 7, 8]]
>>> reshape(seq, (4,)) # tuples of 4
[(1, 2, 3, 4), (5, 6, 7, 8)]
>>> reshape(seq, (2, 2)) # tuples of 4
[(1, 2, 3, 4), (5, 6, 7, 8)]
>>> reshape(seq, (2, [2])) # (i, i, [i, i])
[(1, 2, [3, 4]), (5, 6, [7, 8])]
>>> reshape(seq, ((2,), [2])) # etc....
[((1, 2), [3, 4]), ((5, 6), [7, 8])]
>>> reshape(seq, (1, [2], 1))
[(1, [2, 3], 4), (5, [6, 7], 8)]
>>> reshape(tuple(seq), ([[1], 1, (2,)],))
(([[1], 2, (3, 4)],), ([[5], 6, (7, 8)],))
>>> reshape(tuple(seq), ([1], 1, (2,)))
(([1], 2, (3, 4)), ([5], 6, (7, 8)))
>>> reshape(list(range(12)), [2, [3], set([2]), (1, (3,), 1)])
[[0, 1, [2, 3, 4], set([5, 6]), (7, (8, 9, 10), 11)]]
Left rotates a list x by the number of steps specified in y.
Examples
>>> from sympy.utilities.iterables import rotate_left
>>> a = [0, 1, 2]
>>> rotate_left(a, 1)
[1, 2, 0]
Right rotates a list x by the number of steps specified in y.
Examples
>>> from sympy.utilities.iterables import rotate_right
>>> a = [0, 1, 2]
>>> rotate_right(a, 1)
[2, 0, 1]
Group the sequence into lists in which successive elements all compare the same with the comparison operator, op: op(seq[i + 1], seq[i]) is True from all elements in a run.
Examples
>>> from sympy.utilities.iterables import runs
>>> from operator import ge
>>> runs([0, 1, 2, 2, 1, 4, 3, 2, 2])
[[0, 1, 2], [2], [1, 4], [3], [2], [2]]
>>> runs([0, 1, 2, 2, 1, 4, 3, 2, 2], op=ge)
[[0, 1, 2, 2], [1, 4], [3], [2, 2]]
Sift the sequence, seq into a dictionary according to keyfunc.
OUTPUT: each element in expr is stored in a list keyed to the value of keyfunc for the element.
See also
ordered
Examples
>>> from sympy.utilities import sift
>>> from sympy.abc import x, y
>>> from sympy import sqrt, exp
>>> sift(range(5), lambda x: x % 2)
{0: [0, 2, 4], 1: [1, 3]}
sift() returns a defaultdict() object, so any key that has no matches will give [].
>>> sift([x], lambda x: x.is_commutative)
{True: [x]}
>>> _[False]
[]
Sometimes you won’t know how many keys you will get:
>>> sift([sqrt(x), exp(x), (y**x)**2],
... lambda x: x.as_base_exp()[0])
{E: [exp(x)], x: [sqrt(x)], y: [y**(2*x)]}
If you need to sort the sifted items it might be better to use ordered which can economically apply multiple sort keys to a squence while sorting.
Generates all ksubsets (combinations) from an nelement set, seq.
A ksubset of an nelement set is any subset of length exactly k. The number of ksubsets of an nelement set is given by binomial(n, k), whereas there are 2**n subsets all together. If k is None then all 2**n subsets will be returned from shortest to longest.
Examples
>>> from sympy.utilities.iterables import subsets
subsets(seq, k) will return the n!/k!/(n  k)! ksubsets (combinations) without repetition, i.e. once an item has been removed, it can no longer be “taken”:
>>> list(subsets([1, 2], 2))
[(1, 2)]
>>> list(subsets([1, 2]))
[(), (1,), (2,), (1, 2)]
>>> list(subsets([1, 2, 3], 2))
[(1, 2), (1, 3), (2, 3)]
subsets(seq, k, repetition=True) will return the (n  1 + k)!/k!/(n  1)! combinations with repetition:
>>> list(subsets([1, 2], 2, repetition=True))
[(1, 1), (1, 2), (2, 2)]
If you ask for more items than are in the set you get the empty set unless you allow repetitions:
>>> list(subsets([0, 1], 3, repetition=False))
[]
>>> list(subsets([0, 1], 3, repetition=True))
[(0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1)]
Topological sort of graph’s vertices.
Parameters :  ``graph`` : tuple[list, list[tuple[T, T]]
``key`` : callable[T] (optional)


Examples
Consider a graph:
++ ++ ++
 7 \  5   3 
++ \ ++ ++
 _\___/ ____ _/ 
 / \___/ \ / 
V V V V 
++ ++ 
 11   8  
++ ++ 
  \____ ___/ _ 
 \ \ / / \ 
V \ V V / V V
++ \ ++  ++
 2    9    10 
++  ++  ++
\________/
where vertices are integers. This graph can be encoded using elementary Python’s data structures as follows:
>>> V = [2, 3, 5, 7, 8, 9, 10, 11]
>>> E = [(7, 11), (7, 8), (5, 11), (3, 8), (3, 10),
... (11, 2), (11, 9), (11, 10), (8, 9)]
To compute a topological sort for graph (V, E) issue:
>>> from sympy.utilities.iterables import topological_sort
>>> topological_sort((V, E))
[3, 5, 7, 8, 11, 2, 9, 10]
If specific tie breaking approach is needed, use key parameter:
>>> topological_sort((V, E), key=lambda v: v)
[7, 5, 11, 3, 10, 8, 9, 2]
Only acyclic graphs can be sorted. If the input graph has a cycle, then ValueError will be raised:
>>> topological_sort((V, E + [(10, 7)]))
Traceback (most recent call last):
...
ValueError: cycle detected
Group iter into tuples of length n. Raise an error if the length of iter is not a multiple of n.
Yield unique elements from seq as an iterator. The second parameter result is used internally; it is not necessary to pass anything for this.
Examples
>>> from sympy.utilities.iterables import uniq
>>> dat = [1, 4, 1, 5, 4, 2, 1, 2]
>>> type(uniq(dat)) in (list, tuple)
False
>>> list(uniq(dat))
[1, 4, 5, 2]
>>> list(uniq(x for x in dat))
[1, 4, 5, 2]
>>> list(uniq([[1], [2, 1], [1]]))
[[1], [2, 1]]
Wrapper to necklaces to return a free (unrestricted) necklace.
Returns a generator of the nsized variations of seq (size N). repetition controls whether items in seq can appear more than once;
See also
sympy.core.compatibility.permutations, sympy.core.compatibility.product
Examples
variations(seq, n) will return N! / (N  n)! permutations without repetition of seq’s elements:
>>> from sympy.utilities.iterables import variations
>>> list(variations([1, 2], 2))
[(1, 2), (2, 1)]
variations(seq, n, True) will return the N**n permutations obtained by allowing repetition of elements:
>>> list(variations([1, 2], 2, repetition=True))
[(1, 1), (1, 2), (2, 1), (2, 2)]
If you ask for more items than are in the set you get the empty set unless you allow repetitions:
>>> list(variations([0, 1], 3, repetition=False))
[]
>>> list(variations([0, 1], 3, repetition=True))[:4]
[(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)]