Graded Poset - The Usual Case

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