T_p (no-singleton partitions): {2: 1, 4: 4, 6: 41, 8: 715, 10: 17722}  (A000296: T_2=1, T_4=4, T_6=41, T_8=715, T_10=17722)
Proposition 4 checked for p in {4,6,8}, d in {1,2,3,5}, n <= 12: failures = 0
Theorem 3 checked for p in {2,4,6,8}, d in {1,2,3}, small n: failures = 0
N(d,p) of Theorem 5 (crude, from Proposition 4):
  p=4: d=1: 35, d=2: 146, d=3: 462, d=5: 3562, d=10: 393598, d=20: 3818729001
  p=6: d=1: 30, d=2: 71, d=3: 150, d=5: 581, d=10: 14144, d=20: 7492786
  p=8: d=1: 31, d=2: 61, d=3: 109, d=5: 316, d=10: 4037, d=20: 610829
  p=10: d=1: 34, d=2: 59, d=3: 95, d=5: 235, d=10: 2037, d=20: 145103
