[brute force over A_n] own enumeration; classes = conjugation orbits
n=10: |A_n| = 1814400, 24 classes (orbits 24); class test = orbits: True; (F3): True; row sums: True; sizes sum: True
   a Hall matching for every class C: True; max |T(C)| = 6
   (10.0 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)
