# 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.

## Properties

• The defining condition of an Eulerian poset P can be equivalently stated in terms of its Möbius function:
${\displaystyle \mu _{P}(x,y)=(-1)^{|y|-|x|}{\text{ for all }}x\leq y.}$
${\displaystyle h_{k}=h_{d-k}\,}$
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

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

## References

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