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:
“I heartily wish you, in the plain home-spun style, a great number of happy new years, well employed in forming both your mind and your manners, to be useful and agreeable to yourself, your country, and your friends.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“This nightmare occupied some ten pages of manuscript and wound off with a sermon so destructive of all hope to non-Presbyterians that it took the first prize. This composition was considered to be the very finest effort of the evening.... It may be remarked, in passing, that the number of compositions in which the word “beauteous” was over-fondled, and human experience referred to as “life’s page,” was up to the usual average.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“There is no luck in literary reputation. They who make up the final verdict upon every book are not the partial and noisy readers of the hour when it appears; but a court as of angels, a public not to be bribed, not to be entreated, and not to be overawed, decides upon every man’s title to fame. Only those books come down which deserve to last.”
—Ralph Waldo Emerson (1803–1882)
“There is nothing on earth more exquisite than a bonny book, with well-placed columns of rich black writing in beautiful borders, and illuminated pictures cunningly inset. But nowadays, instead of looking at books, people read them. A book might as well be one of those orders for bacon and bran.”
—George Bernard Shaw (1856–1950)