Eulerian poset

Examples

• The face lattice of a convex polytope, consisting of its faces, together with the smallest element, the empty face, and the largest element, the polytope itself, is an Eulerian lattice. The odd–even condition follows from Euler's formula.
• Any simplicial generalized homology sphere is an Eulerian lattice.
• Let L be a regular cell complex such that |L| is a manifold with the same Euler characteristic as the sphere of the same dimension. Then the poset of cells of L, ordered by the inclusion of their closures, is Eulerian.
• Let W be a Coxeter group with Bruhat order. Then is an Eulerian poset.

  Properties

• The defining condition of an Eulerian poset P can be equivalently stated in terms of its Möbius function:
• The dual of an Eulerian poset, obtained by reversing the partial order, is Eulerian.
• Richard Stanley defined the toric h-vector of a ranked poset, which generalizes the h-vector of a simplicial polytope. He proved that the Dehn–Sommerville equations