Eulerian poset

In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler. Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics, such as various restrictions on f-vectors of convex simplicial polytopes, to this more general setting.

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 (this condition is vacuous if the dimension is odd). 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 (W,≤) is an Eulerian poset.

Properties

The defining condition of an Eulerian poset P can be equivalently stated in terms of its Möbius function:

\mu _{P}(x,y)=(-1)^{|y|-|x|}{\text{ for all }}x\leq y.

The dual of an Eulerian poset with a top element, 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.[1] He proved that the Dehn–Sommerville equations

h_{k}=h_{d-k}\,

hold for an arbitrary Eulerian poset of rank d + 1.[2] However, for an Eulerian poset arising from a regular cell complex or a convex polytope, the toric h-vector neither determines, nor is neither determined by the numbers of the cells or faces of different dimension and the toric h-vector does not have a direct combinatorial interpretation.

Notes

Enumerative combinatorics, 3.14, p. 138; formerly called the generalized h-vector.
Enumerative combinatorics, Theorem 3.14.9

References

Richard P. Stanley, Enumerative Combinatorics, Volume 1. Cambridge University Press, 1997 ISBN 0-521-55309-1

Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Ordered vector space Partially ordered Positive cone Riesz space Upper set Young's lattice

Eulerian poset

Examples

Properties

Notes

References

See also