[brute force over A_n] own enumeration; classes = conjugation orbits
n=4: |A_n| = 12, 4 classes (orbits 4); class test = orbits: True; (F3): True; row sums: True; sizes sum: True
   NO Hall matching for C in [((3, 1), 1), ((3, 1), -1)]; max |T(C)| = 4
   A_4, S = V_4 (2 classes), C = ((3, 1), 1): SC consists of 1 class(es): [((3, 1), 1)]
   A_4, S = V_4 (2 classes), C = ((3, 1), -1): SC consists of 1 class(es): [((3, 1), -1)]
   (0.0 s)
n=5: |A_n| = 60, 5 classes (orbits 5); class test = orbits: True; (F3): True; row sums: True; sizes sum: True
   a Hall matching for every class C: True; max |T(C)| = 3
   (0.0 s)
n=6: |A_n| = 360, 7 classes (orbits 7); class test = orbits: True; (F3): True; row sums: True; sizes sum: True
   a Hall matching for every class C: True; max |T(C)| = 3
   (0.0 s)
n=7: |A_n| = 2520, 9 classes (orbits 9); class test = orbits: True; (F3): True; row sums: True; sizes sum: True
   a Hall matching for every class C: True; max |T(C)| = 3
   (0.0 s)
n=8: |A_n| = 20160, 14 classes (orbits 14); class test = orbits: True; (F3): True; row sums: True; sizes sum: True
   a Hall matching for every class C: True; max |T(C)| = 5
   (0.1 s)
n=9: |A_n| = 181440, 18 classes (orbits 18); class test = orbits: True; (F3): True; row sums: True; sizes sum: True
   a Hall matching for every class C: True; max |T(C)| = 6
   (0.8 s)
[Proposition 1.3(a)]
   S_3: SC = S_n minus A_n: True; k(S) = 2 > k(SC) = 1: True
   S_4: SC = S_n minus A_n: True; k(S) = 3 > k(SC) = 2: True
   S_5: SC = S_n minus A_n: True; k(S) = 4 > k(SC) = 3: True
   S_6: SC = S_n minus A_n: True; k(S) = 6 > k(SC) = 5: True
   S_7: SC = S_n minus A_n: True; k(S) = 8 > k(SC) = 7: True
   3 <= n <= 200: #even - #odd partitions = #partitions into distinct odd parts >= 1: True (dynamic programming; agrees with direct enumeration for n <= 30: True)
