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Add derangements() recipe (gh-143671)

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Raymond Hettinger 2026-01-10 19:47:27 -06:00 committed by GitHub
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Doc/library/itertools.rst
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@ -842,7 +842,7 @@ and :term:`generators <generator>` which incur interpreter overhead.
from contextlib import suppress
from functools import reduce
from math import comb, prod, sumprod, isqrt
from operator import itemgetter, getitem, mul, neg
from operator import is_not, itemgetter, getitem, mul, neg
def take(n, iterable):
"Return first n items of the iterable as a list."
@ -978,6 +978,16 @@ and :term:`generators <generator>` which incur interpreter overhead.
slices = starmap(slice, combinations(range(len(seq) + 1), 2))
return map(getitem, repeat(seq), slices)
def derangements(iterable, r=None):
"Produce r length permutations without fixed points."
# derangements('ABCD') → BADC BCDA BDAC CADB CDAB CDBA DABC DCAB DCBA
# Algorithm credited to Stefan Pochmann
seq = tuple(iterable)
pos = tuple(range(len(seq)))
have_moved = map(map, repeat(is_not), repeat(pos), permutations(pos, r=r))
valid_derangements = map(all, have_moved)
return compress(permutations(seq, r=r), valid_derangements)
def iter_index(iterable, value, start=0, stop=None):
"Return indices where a value occurs in a sequence or iterable."
# iter_index('AABCADEAF', 'A') → 0 1 4 7
@ -1663,6 +1673,36 @@ The following recipes have a more mathematical flavor:
['A', 'AB', 'ABC', 'ABCD', 'B', 'BC', 'BCD', 'C', 'CD', 'D']
>>> ' '.join(map(''.join, derangements('ABCD')))
'BADC BCDA BDAC CADB CDAB CDBA DABC DCAB DCBA'
>>> ' '.join(map(''.join, derangements('ABCD', 3)))
'BAD BCA BCD BDA CAB CAD CDA CDB DAB DCA DCB'
>>> ' '.join(map(''.join, derangements('ABCD', 2)))
'BA BC BD CA CD DA DC'
>>> ' '.join(map(''.join, derangements('ABCD', 1)))
'B C D'
>>> ' '.join(map(''.join, derangements('ABCD', 0)))
''
>>> # Compare number of derangements to https://oeis.org/A000166
>>> [len(list(derangements(range(n)))) for n in range(10)]
[1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496]
>>> # Verify that identical objects are treated as unique by position
>>> identical = 'X'
>>> distinct = 'x'
>>> seq1 = ('A', identical, 'B', identical)
>>> result1 = ' '.join(map(''.join, derangements(seq1)))
>>> result1
'XAXB XBXA XXAB BAXX BXAX BXXA XAXB XBAX XBXA'
>>> seq2 = ('A', identical, 'B', distinct)
>>> result2 = ' '.join(map(''.join, derangements(seq2)))
>>> result2
'XAxB XBxA XxAB BAxX BxAX BxXA xAXB xBAX xBXA'
>>> result1 == result2
False
>>> result1.casefold() == result2.casefold()
True
>>> list(powerset([1,2,3]))
[(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]
>>> all(len(list(powerset(range(n)))) == 2**n for n in range(18))