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* 2^{n2/4+3n/2} e^{n}*n*^{–n–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).

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send feedback and suggestions to bombelli at olemiss.edu – modified 18 jan 2016