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In [[combinatorics|combinatorial]] mathematics, an '''Eulerian poset''' is a [[graded poset]] in which every nontrivial [[Partially ordered set#Interval|interval]] has the same number of elements of even rank as of odd rank. An Eulerian poset which is a [[lattice (order)|lattice]] is an '''Eulerian lattice'''. These objects are named after [[Leonhard Euler]]. Eulerian lattices generalize [[face lattice]]s of [[convex polytope]]s and much recent research has been devoted to extending known results from [[polyhedral combinatorics]], such as various restrictions on ''f''-vectors of convex [[simplicial polytope]]s, 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 characteristic|Euler's formula]].

* Any [[simplicial sphere|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 [[n-sphere|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 [[Incidence algebra|Möbius function]]:

:: <math> \mu_P(x,y)=(-1)^{|y|-|x|} \text{ for all } x\leq y.</math>

* The dual of an Eulerian poset, obtained by reversing the partial order, is Eulerian.

* [[Richard P. Stanley|Richard Stanley]] defined the '''toric ''h''-vector''' of a [[ranked poset]], which generalizes the [[h-vector|''h''-vector]] of a simplicial polytope.<ref>''Enumerative combinatorics'', 3.14, p. 138; formerly called the ''generalized'' ''h''-vector''.</ref> He proved that the [[Dehn–Sommerville equations]]

:: <math> h_k = h_{d-k} \, </math>

: hold for an arbitrary Eulerian poset of rank ''d'' + 1.<ref>''Enumerative combinatorics'', Theorem 3.14.9</ref> 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 ==
{{reflist}}

== References ==

* [[Richard P. Stanley]], [http://www-math.mit.edu/~rstan/ec/ ''Enumerative Combinatorics''], Volume 1. Cambridge University Press, 1997 ISBN 0-521-55309-1

==See also==
*[[Abstract polytope]]

[[Category:Algebraic combinatorics]]

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