For the history of this subject you can see records in combinatorics and number theory.
For the precise table of the computation of the number of posets with 12 or 14 elements see P(12) or P(14).

REFERENCES

• Calculs exhaustifs sur les posets d’au plus 7 éléments.
  SINGULARITE, vol.2 n4 p.10-24, avril 1991. (N.Lygeros)

• Petits posets : dénombrement, représentabilité par cercles et
  compenseurs.
  C.R.Acad.Sci.Paris, t.313, s.I, p.417-420, septembre 1991. (R.Fraïssé, N.Lygeros)

• The number of orders with thirteen elements.
  Order, vol.9 p.203-204, 1992 (C.Chaunier, N.Lygeros)

  Relations	P(13,r)
  r=0	1
  r=1	1
  r=2	3
  r=3	7
  r=4	19
  r=5	47
  r=6	133
  r=7	354
  r=8	1014
  r=9	2874
  r=10	8305
  r=11	23513
  r=12	65215
  r=13	173481
  r=14	441249
  r=15	1062532
  r=16	2419194
  r=17	5194267
  r=18	10529510
  r=19	20169973
  r=20	36606102
  r=21	63090851
  r=22	103573457
  r=23	162384152
  r=24	243809985
  r=25	351390204
  r=26	487237576
  r=27	651206672
  r=28	840404152
  r=29	1048785819
  r=30	1267416540
  r=31	1484925018
  r=32	1688672630
  r=33	1865878896
  r=34	2005172954
  r=35	2097659160
  r=36	2138021170
  r=37	2124818344
  r=38	2060635454
  r=39	1951423800
  r=40	1805816407
  r=41	1633935577
  r=42	1446444433
  r=43	1253457366
  r=44	1063880995
  r=45	884825225
  r=46	721452090
  r=47	576924933
  r=48	452654555
  r=49	348576046
  r=50	263545083
  r=51	195684307
  r=52	142728742
  r=53	102283393
  r=54	72028601
  r=55	49849120
  r=56	33906587
  r=57	22666616
  r=58	14891283
  r=59	9613263
  r=60	6096379
  r=61	3796733
  r=62	2320757
  r=63	1391478
  r=64	817624
  r=65	470396
  r=66	264558
  r=67	145258
  r=68	77647
  r=69	40260
  r=70	20165
  r=71	9660
  r=72	4391
  r=73	1862
  r=74	714
  r=75	241
  r=76	66
  r=77	12
  r=78	1

   

  These results confirm R. Fraïssé’s conjecture on unimodality.