Graded poset

[1][2] Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram.

This covers many finite cases of interest; see picture for a negative example.

In some common posets such as the face lattice of a convex polytope there is a natural grading by dimension, which if used as rank function would give the minimal element, the empty face, rank −1.

Allowing arbitrary integers as rank would however give a fundamentally different notion; for instance the existence of a minimal element would no longer be assured.

For instance, the integers (with the usual order) cannot be a graded poset, nor can any interval (with more than one element) of rational or real numbers.

It also means that (for positive integer rank functions) compatibility of ρ with the ordering follows from the requirement about covers.

As a variant of the definition of a graded poset, Birkhoff[7] allows rank functions to have arbitrary (rather than only nonnegative) integer values.

In this variant, the integers can be graded (by the identity function) in his setting, and the compatibility of ranks with the ordering is not redundant.

As a third variant, Brightwell and West[8] define a rank function to be integer-valued, but don't require its compatibility with the ordering; hence this variant can grade even e.g. the real numbers by any function, as the requirement about covers is vacuous for this example.

This condition is necessary since every step in a maximal chain is a covering relation, which should change the rank by 1.

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.

The Whitney numbers are connected with a lot of important combinatorial theorems.