The Usual Case
Many authors in combinatorics define graded posets in such a way that all minimal elements of P must have rank 0, and moreover that there is a maximal rank r which is the rank of any maximal element. Then being graded means that all maximal chains have length r, as is indicated above. In this case one says that P has rank r.
Furthermore, in this case with the rank levels are associated the rank numbers or Whitney numbers . These numbers are defined by = number of elements of P having rank i .
The Whitney numbers are connected with a lot of important combinatorial theorems. The classic example is Sperner's theorem which can be formulated as follows:
- For the powerset of every finite set the maximum cardinality of a Sperner family equals the maximum Whitney number.
This means:
- Every finite powerset has the Sperner property
Read more about this topic: Graded Poset
Famous quotes containing the words usual and/or case:
“America is the only nation in history which miraculously has gone directly from barbarism to degeneration without the usual interval of civilization.”
—Georges Clemenceau (1841–1929)
“I love to weigh, to settle, to gravitate toward that which most strongly and rightfully attracts me;Mnot hang by the beam of the scale and try to weigh less,—not suppose a case, but take the case that is; to travel the only path I can, and that on which no power can resist me. It affords me no satisfaction to commence to spring an arch before I have got a solid foundation.”
—Henry David Thoreau (1817–1862)