Eulerian poset

A graded poset P is Eulerian if for any xy in P, the Möbius function ( of the interval [x,y] is given by


where ρ is the rank function of P. If P is finite and graded with strictly positive rank, then we can equivalently and more simply say that P is Eulerian if and only if each nontrivial interval of P has the same number of elements of even rank as odd rank.

For example, the latticeMathworldPlanetmath ( M3 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 Bn is always Eulerian.


Each interval of Bn is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to a smaller Boolean lattice, so we can give a proof by induction that Bn is Eulerian. Now B0 is a single point and so there is only one interval, which has Möbius function 1. So to show that Bn is Eulerian, we just need to show that the Möbius function of Bn is (-1)n. Recall that by definition,

μBn(Bn)=-S{1,,n}μ(,S) (*)

Thus S{1,,n}μBn(,S)=0. We can apply the binomial theoremMathworldPlanetmath to 0 to see that 0=(-1+1)n=k=0n(nk)(-1)k and thus

(-1)n =-k=0n-1(nk)(-1)k
=-S{1,,n}μ(,S) (**)

It then follows from Equation (*) and Equation (**) that μ(Bn)=(-1)n.

Eulerian posets are generalizationsPlanetmathPlanetmath of certain special posets, such as face lattices of polytopes.

Eulerian posets of small rank

In this sectionPlanetmathPlanetmath we classify bounded Eulerian posets of rank less than 4.

For n<2, there is exactly one bounded poset of rank n, the chain, and it is Eulerian.

A bounded poset of rank 2 is determined precisely by its number of coatoms. The ab-index of a rank 2 bounded Eulerian poset with r coatoms is a+(r-1)b, which is a cd-index if and only if r=2. Since every Eulerian poset has a cd-index, this implies that every rank 2 bounded Eulerian poset has exactly two coatoms. Thus up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmath B2 is the only rank 2 bounded Eulerian poset.

Let P be a rank 3 bounded Eulerian poset. Since P has as many elements of odd rank as even rank, it has exactly the same number of atoms as coatoms. Since every rank 2 interval in P is a copy of B2, it follows that P is not a chain and so has n2 atoms. Moreover, every atom is covered by exactly two coatoms, and every coatom covers exactly two atoms. That is, the Hasse diagramMathworldPlanetmath of the interior of P is a 2-regular bipartite graphMathworldPlanetmath. 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 P consisting of the points on a cycle of length 2k plus the minimum and maximum elements of P is the face poset of a k-gon. Hence P itself is the face poset of the disjoint unionMathworldPlanetmath of one or more k-gons for various k2.


Title Eulerian poset
Canonical name EulerianPoset
Date of creation 2013-03-22 14:08:16
Last modified on 2013-03-22 14:08:16
Owner mps (409)
Last modified by mps (409)
Numerical id 9
Author mps (409)
Entry type Definition
Classification msc 06A11
Classification msc 06A07
Related topic GradedPoset