review

src/sage/combinat/tableau.py

@@ -5650,10 +5650,11 @@ def random_element(self):
         # We compute the number of involutions of size ``size``.
         from sage.misc.prandom import randrange
         involution_index = randrange(0, StandardTableaux(self.size).cardinality())
+        # ``involution_index`` is our random integer `r`.
         partial_sum = 0
         fixed_point_number = self.size % 2
-
-        from sage.functions.other import binomial
+        # ``fixed_point_number`` will become `k`.
+        from sage.rings.arith import binomial
         while True:
             # We add the number of involutions with ``fixed_point_number``
             # fixed points.
@@ -5668,10 +5669,12 @@ def random_element(self):
         # We generate a subset of size "fixed_point_number" of the set {1,
         # ..., size}.
         from sage.misc.prandom import sample
-        fixed_point_positions = set(sample(range(1, self.size + 1), fixed_point_number))
-        # We generate a list of tuples with the first part being the fixed
-        # points of the permutation and the second part being a perfect
-        # matching on the remaining values.
+        fixed_point_positions = set(sample(xrange(1, self.size + 1), fixed_point_number))
+        # We generate a list of tuples which will form the cycle
+        # decomposition of our random involution. This list contains
+        # singletons (corresponding to the fixed points of the
+        # involution) and pairs (forming a perfect matching on the
+        # remaining values).
         from sage.combinat.perfect_matching import PerfectMatchings
         permutation_cycle_rep = [(fixed_point,) for fixed_point in fixed_point_positions] + \
                                 list(PerfectMatchings(set(range(1, self.size + 1)) - set(fixed_point_positions)).random_element())