Number of Partial Orders
Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:
| Number of n-element binary relations of different types | ||||||||
|---|---|---|---|---|---|---|---|---|
| n | all | transitive | reflexive | preorder | partial order | total preorder | total order | equivalence relation |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 16 | 13 | 4 | 4 | 3 | 3 | 2 | 2 |
| 3 | 512 | 171 | 64 | 29 | 19 | 13 | 6 | 5 |
| 4 | 65536 | 3994 | 4096 | 355 | 219 | 75 | 24 | 15 |
| OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |
The number of strict partial orders is the same as that of partial orders.
If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (sequence A000112 in OEIS).
Read more about this topic: Partially Ordered Set
Famous quotes containing the words number of, number, partial and/or orders:
“My tendency to nervousness in my younger days, in view of the fact of a number of near relatives on both my father’s and mother’s side of the house having become insane, gave some serious uneasiness. I made up my mind to overcome it.... In the cross-examination of witnesses before a crowded court-house ... I soon found I could control myself even in the worst of testing cases. Finally, in battle.”
—Rutherford Birchard Hayes (1822–1893)
“To make life more bearable and pleasant for everybody, choose the issues that are significant enough to fight over, and ignore or use distraction for those you can let slide that day. Picking your battles will eliminate a number of conflicts, and yet will still leave you feeling in control.”
—Lawrence Balter (20th century)
“And meanwhile we have gone on living,
Living and partly living,
Picking together the pieces,
Gathering faggots at nightfall,
Building a partial shelter,
For sleeping and eating and drinking and laughter.”
—T.S. (Thomas Stearns)
“One cannot be a good historian of the outward, visible world without giving some thought to the hidden, private life of ordinary people; and on the other hand one cannot be a good historian of this inner life without taking into account outward events where these are relevant. They are two orders of fact which reflect each other, which are always linked and which sometimes provoke each other.”
—Victor Hugo (1802–1885)