by 1. The condition is also sufficient, since when it holds, one can use the mentioned length to define the rank of x (the length of a finite chain is its number of "steps", so one less than its number of elements), and whenever x covers y, adjoining x to a maximal chain in [Ô, y] gives a maximal chain in [Ô, x]. If P also has a greatest element Î (so that it is a bounded poset), then the previous condition can be simplified to the requirement that all maximal chains in P have the same (finite) length. This suffices, since any pair of maximal chains in [Ô, x] can be extended by a maximal chain in [x, Î] to give a pair of maximal chains in P. Note Stanley defines a poset to be graded of length n if all its maximal chains have length n (Stanley 1997, p.99). This definition is given in a context where interest is mostly in finite posets, and although the book subsequently often drops the part "of length n", it does not seem appropriate to use this as definition of "graded" for general posets, because (1) it says nothing about posets whose maximal chains are infinite, in particular (2) it excludes important posets like Young's lattice. Also it is not clear why in a graded poset all minimal elements, as well as all maximal elements, should be required to have the same length, even if Stanley gives examples making clear that he does mean to require that (ibid, pp.216 and 219). 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 that 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, to 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 power set of every finite set the maximum cardinality of a Sperner family equals the maximum Whitney