Add derangements() recipe

Doc/library/itertools.rst

@@ -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))