where is the rank function of . If is finite and graded with strictly positive rank, then we can equivalently and more simply say that is Eulerian if and only if each nontrivial interval of has the same number of elements of even rank as odd rank.
For example, the lattice (http://planetmath.org/Lattice) is not Eulerian, because it has 3 elements of rank 1 but only 2 elements of ranks 0 or 2.
On the other hand, the Boolean lattice is always Eulerian.
Each interval of is isomorphic to a smaller Boolean lattice, so we can give a proof by induction that is Eulerian. Now is a single point and so there is only one interval, which has Möbius function 1. So to show that is Eulerian, we just need to show that the Möbius function of is . Recall that by definition,
Thus . We can apply the binomial theorem to to see that and thus
It then follows from Equation and Equation that .
Eulerian posets of small rank
For , there is exactly one bounded poset of rank , the chain, and it is Eulerian.
A bounded poset of rank is determined precisely by its number of coatoms. The ab-index of a rank bounded Eulerian poset with coatoms is , which is a cd-index if and only if . Since every Eulerian poset has a cd-index, this implies that every rank bounded Eulerian poset has exactly two coatoms. Thus up to isomorphism is the only rank bounded Eulerian poset.
Let be a rank bounded Eulerian poset. Since has as many elements of odd rank as even rank, it has exactly the same number of atoms as coatoms. Since every rank interval in is a copy of , it follows that is not a chain and so has atoms. Moreover, every atom is covered by exactly two coatoms, and every coatom covers exactly two atoms. That is, the Hasse diagram of the interior of is a 2-regular bipartite graph. This graph admits a decomposition into even cycles. In each cycle, half of the nodes are atoms and half are coatoms. The non-induced subposet of consisting of the points on a cycle of length plus the minimum and maximum elements of is the face poset of a -gon. Hence itself is the face poset of the disjoint union of one or more -gons for various .
- 1 Stanley, R., Enumerative Combinatorics, vol. 1, 2nd ed., Cambridge University Press, Cambridge, 1996.
|Date of creation||2013-03-22 14:08:16|
|Last modified on||2013-03-22 14:08:16|
|Last modified by||mps (409)|