sagemathgh-41024: fix cardinality of Permutations_setk
    
Fixes sagemath#41022
    
URL: sagemath#41024
Reported by: Martin Rubey
Reviewer(s): mathzeta

Author: Release Manager

src/sage/combinat/permutation.py

@@ -243,7 +243,7 @@
 import operator
 from typing import TYPE_CHECKING
 
-from sage.arith.misc import factorial, multinomial
+from sage.arith.misc import factorial, multinomial, falling_factorial
 from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
 from sage.categories.finite_permutation_groups import FinitePermutationGroups
 from sage.categories.finite_weyl_groups import FiniteWeylGroups
@@ -2144,7 +2144,7 @@ def ishift(self, i):
         Return the ``i``-shift of ``self``. If an ``i``-shift of ``self``
         can't be performed, then ``self`` is returned.
 
-        An `i`-shift can be applied when `i` is not inbetween `i-1` and
+        An `i`-shift can be applied when `i` is not between `i-1` and
         `i+1`. The `i`-shift moves `i` to the other side, and leaves the
         relative positions of `i-1` and `i+1` in place. All other entries
         of the permutations are also left in place.
@@ -6243,7 +6243,7 @@ def __init__(self, n, k):
         """
         TESTS::
 
-            sage: P = Permutations(3,2)
+            sage: P = Permutations(5, 3)
             sage: TestSuite(P).run()
         """
         self.n = ZZ(n)
@@ -6320,6 +6320,8 @@ def __iter__(self) -> Iterator[Permutation]:
 
     def cardinality(self) -> Integer:
         """
+        Return the cardinality of the set.
+
         EXAMPLES::
 
             sage: Permutations(3,0).cardinality()
@@ -6334,7 +6336,7 @@ def cardinality(self) -> Integer:
             0
         """
         if 0 <= self._k <= self.n:
-            return factorial(self.n) // factorial(self.n - self._k)
+            return falling_factorial(self.n, self._k)
         return ZZ.zero()
 
     def random_element(self):
@@ -6409,7 +6411,7 @@ def __init__(self, mset):
         """
         TESTS::
 
-            sage: S = Permutations(['c','a','c'])
+            sage: S = Permutations(['c','a','c','d'])
             sage: TestSuite(S).run()
         """
         self.mset = mset
@@ -6898,7 +6900,7 @@ def __init__(self, mset, k):
         """
         TESTS::
 
-            sage: P = Permutations([1,2,2],2)
+            sage: P = Permutations([1,2,2,3], 2)
             sage: TestSuite(P).run()                                                    # needs sage.libs.gap
         """
         Permutations_mset.__init__(self, mset)
@@ -7014,12 +7016,26 @@ def __init__(self, s, k):
         """
         TESTS::
 
-            sage: P = Permutations([1,2,4],2)
+            sage: P = Permutations([1,2,4,5], 2)
             sage: TestSuite(P).run()
         """
         Permutations_set.__init__(self, s)
         self._k = k
 
+    def cardinality(self) -> Integer:
+        """
+        Return the cardinality of the set.
+
+        EXAMPLES::
+
+            sage: Permutations([1,2,4,5], 2).cardinality()
+            12
+        """
+        n = len(self._set)
+        if 0 <= self._k <= n:
+            return falling_factorial(n, self._k)
+        return ZZ.zero()
+
     def __contains__(self, x):
         """
         EXAMPLES::
@@ -9841,6 +9857,8 @@ def __init__(self, n):
 
     def cardinality(self):
         """
+        Return the cardinality of the set.
+
         EXAMPLES::
 
             sage: Permutations(5, avoiding=[1, 3, 2]).cardinality()
@@ -9918,6 +9936,8 @@ def __init__(self, n):
 
     def cardinality(self) -> Integer:
         """
+        Return the cardinality of the set.
+
         EXAMPLES::
 
             sage: Permutations(5, avoiding=[1, 2, 3]).cardinality()
@@ -9990,6 +10010,8 @@ def __init__(self, n):
 
     def cardinality(self):
         """
+        Return the cardinality of the set.
+
         EXAMPLES::
 
             sage: Permutations(5, avoiding=[3, 2, 1]).cardinality()
@@ -10022,6 +10044,8 @@ def __init__(self, n):
 
     def cardinality(self):
         """
+        Return the cardinality of the set.
+
         EXAMPLES::
 
             sage: Permutations(5, avoiding=[2, 3, 1]).cardinality()
@@ -10054,6 +10078,8 @@ def __init__(self, n):
 
     def cardinality(self):
         """
+        Return the cardinality of the set.
+
         EXAMPLES::
 
             sage: Permutations(5, avoiding=[3, 1, 2]).cardinality()
@@ -10086,6 +10112,8 @@ def __init__(self, n):
 
     def cardinality(self):
         """
+        Return the cardinality of the set.
+
         EXAMPLES::
 
             sage: Permutations(5, avoiding=[2, 1, 3]).cardinality()