Enumeration and Properties of Partially Ordered Sets

In General up to posets.
* Symbols: We denote by $$\cal L$$n the set of labelled n-element posets, by $$\cal P$$n the set of unlabelled ones, and by $$\cal C$$n the set of connected ones.
* Status: As of 2000, exact numbers are known up to |$$\cal P$$14| = 1.338.193.159.771 and |$$\cal L$$16| (their values are not close to those given by the n → ∞ formula, because 16 is still too small); By 2012, numbers up to |$$\cal P$$16| and |$$\cal L$$18| were known (see the lists below).
* Results: The number of labelled, unlabelled, and connected n-element posets, for the first values of n is

 n |$$\cal L$$n| |$$\cal P$$n| |$$\cal C$$n| 1 1 1 1 2 3 2 1 3 19 5 3 4 219 16 10 5 4231 63 44 6 130,023 318 238 7 6,129,859 2,045 1,650 8 431,723,379 16,999 14,512 9 44,511,042,511 183,231 163,341 10 6,611,065,248,783 2,567,284 2,360,719 11 1,396,281,677,105,899 46,749,427 43,944,974 12 414,864,951,055,853,499 1,104,891,746 1,055,019,099 13 171,850,728,381,587,059,351 33,823,827,452 32,664,984,238 14 98,484,324,257,128,207,032,183 1,338,193,159,771 1,303,143,553,205 15 77,567,171,020,440,688,353,049,939 68,275,077,901,156 66,900,392,672,168 16 83,480,529,785,490,157,813,844,256,579 4,483,130,665,195,087 4,413,439,778,321,689 17 122,152,541,250,295,322,862,941,281,269,151 18 241,939,392,597,201,176,602,897,820,148,085,023

* Examples: The first few sets, for n up to 3, are $$\cal P$$1 = {$$\bullet$$,}, $$\cal P$$2 = {$$\bullet\,\bullet$$, }, $$\cal P$$3 = {$$\bullet\bullet\bullet$$, $$\bullet$$ , , , }, while $$\cal C$$1 = {$$\bullet$$}, $$\cal C$$2 = {}, $$\cal C$$3 = {, , }.
@ General references: Culberson & Rawlins Ord(90) [up to n = 11]; Erné & Stege Ord(91) [up to n = 14]; Chaunier & Lygeros Ord(92) [n = 13]; Heitzig & Reinhold Ord(00), Lygeros & Zimmermann www(00) [n = 14]; Brinkmann & McKay Ord(02) [n = 15, 16].
@ Special types of posets: Lewis & Zhang JCTA(13)-a1106 [(3+1)-avoiding posets]; Stanley PAMS(74) [posets generated by disjoint unions and ordinal sums].
> Online resources: The Online Encyclopedia of Integer Sequences (OEIS) site; Chapel Hill Poset Atlas site.

Asymptotic Properties
* Asymptotic numbers: For n → ∞, the asymptotic value for the number of labelled posets in the case of even n is

|$$\cal P$$n| ~ C 2n2/4+3n/2 en nn–1,   where   C ≅ 0.8059,

and something quite similar for n odd.
@ General references: Kleitman & Rothschild TAMS(75); Dhar PJM(80)-mr; Henson et al a1504 [onset of the Kleitman-Rothschild 3-layer structure].
@ Phase transitions: Dhar JMP(78); Kleitman & Rothschild PhyA(79); Pittel & Tungol RSA(01).