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)
“In the end we beat them with Levi 501 jeans. Seventy-two years of Communist indoctrination and propaganda was drowned out by a three-ounce Sony Walkman. A huge totalitarian system ... has been brought to its knees because nobody wants to wear Bulgarian shoes.... Now they’re lunch, and we’re number one on the planet.”
—P.J. (Patrick Jake)
“We were soon in the smooth water of the Quakish Lake,... and we had our first, but a partial view of Ktaadn, its summit veiled in clouds, like a dark isthmus in that quarter, connecting the heavens with the earth.”
—Henry David Thoreau (1817–1862)
“God is a foreman with certain definite views
Who orders life in shifts of work and leisure.”
—Seamus Heaney (b. 1939)