For the history of this subject see records in combinatorics and number theory.
For the precise table of the computation of the number of posets with 12 or 13 elements see P(12) or P(13).

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)
• Le nombre de posets à isomorphie près ayant 12 éléments.
  Theoretical Computer Science, n123 p.89-94, février 1994. (C.Chaunier, N.Lygeros)HELPERSRobert Erra, Bernard Landreau, John Lygeros, Jon Kierkegaard, Ilias Kotsireas, Thomas Papanikolaou, Robert Piche.PARTIAL RESULTS:

  Relations	P(14,r)	Name	file
  r=0	1	–	–
  r=1	1	–	–
  r=2	3	–	–
  r=3	7	–	–
  r=4	19	–	–
  r=5	47	–	–
  r=6	133	–	–
  r=7	355	–	–
  r=8	1020	–	–
  r=9	2921	–	–
  r=10	8632	–	–
  r=11	25486	–	–
  r=12	75366	–	–
  r=13	217990	–	–
  r=14	613701	–	–
  r=15	1659796	–	–
  r=16	4290662	–	–
  r=17	10541968	–	–
  r=18	24574810	–	–
  r=19	54282974	–	–
  r=20	113695015	–	–
  r=21	226043606	–	–
  r=22	427434085	–	–
  r=23	770364708	N. Lygeros	–
  r=24	1326574992	P. Zimmermann	–
  r=25	2187816296	Ilias Kotsireas	–
  r=26	3463953205	Jon Kierkegaard	res26
  r=27	5276878418	Jon Kierkegaard	res27
  r=28	7750426625	Robert Erra, N. Lygeros	res28
  r=29	10995675430	P. Zimmermann	–
  r=30	15093556839	Ilias Kotsireas, N. Lygeros	res30
  r=31	20075844431	Robert Piche, N. Lygeros	res31
  r=32	25908418088	Robert Piche	–
  r=33	32478923331	Robert Piche	–
  r=34	39592931224	Robert Piche	–
  r=35	46979502004	Robert Piche	res35
  r=36	54307727007	N. Lygeros	res36
  r=37	61211924728	Bernard Landreau	–
  r=38	67323642504	Thomas Papanikolaou	res38
  r=39	72305442194	Jon Kierkegaard, P. Zimmermann	res39
  r=40	75882802702	Jon Kierkegaard, P. Zimmermann	res40
  r=41	77869195951	Bernard Landreau	res41
  r=42	78181684187	Jon Kierkegaard, P. Zimmermann	res42
  r=43	76844939859	P. Zimmermann	res43
  r=44	73984081629	N. Lygeros	res44
  r=45	69807866652	Robert Piche, P. Zimmermann	res45
  r=46	64585100397	Jon Kierkegaard, N. Lygeros	res46
  r=47	58617924003	Jon Kierkegaard, P. Zimmermann	res47
  r=48	52215061893	N. Lygeros	res48
  r=49	45668403312	Bernard Landreau	res49
  r=50	39234427091	Jon Kierkegaard	res50
  r=51	33122134980	Jon Kierkegaard, N. Lygeros	res51
  r=52	27486947185	Jon Kierkegaard	res52
  r=53	22430638497	Jon Kierkegaard	res53
  r=54	18005406984	Jon Kierkegaard	res54
  r=55	14221396017	Jon Kierkegaard	res55
  r=56	11055548408	Jon Kierkegaard	res56
  r=57	8461181255	John Lygeros	res57
  r=58	6376703312	Robert Erra	res58
  r=59	4733381232	Jon Kierkegaard	res59
  r=60	3461298437	N. Lygeros	–
  r=61	2493875402	–	–
  r=62	1770675277	–	–
  r=63	1239039207	–	–
  r=64	854569812	–	–
  r=65	580970065	–	–
  r=66	389321463	–	–
  r=67	257162269	–	–
  r=68	167425100	–	–
  r=69	107425128	–	–
  r=70	67918114	–	–
  r=71	42302121	–	–
  r=72	25947267	–	–
  r=73	15667333	–	–
  r=74	9307819	–	–
  r=75	5436986	–	–
  r=76	3120222	–	–
  r=77	1757459	–	–
  r=78	970342	–	–
  r=79	524242	–	–
  r=80	276596	–	–
  r=81	142020	–	–
  r=82	70684	–	–
  r=83	33907	–	–
  r=84	15527	–	–
  r=85	6724	–	–
  r=86	2701	–	–
  r=87	978	–	–
  r=88	309	–	–
  r=89	78	–	–
  r=90	13	–	–
  r=91	1	–	–