Flag Vectors of Eulerian Partially Ordered Sets

Abstract

The closed cone of flag vectors of Eulerian partially ordered sets is studied. A new family of linear inequalities valid for Eulerian flag vectors is given. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise to extreme rays of the cone for Eulerian posets. Other extreme posets are formed from consideration of the cd-index. The cone of Eulerian flag vectors is completely determined up through rank seven.

Flag Enumeration in Polytopes, Eulerian Partially Ordered Sets and Coxeter Groups

“…The cones of all flag vectors are known for Eulerian posets through rank 6. The best references for this are [8,9]. For S ⊆ [n] let the flag h-vector be defined by…”

“…Formulas expressing these in terms of the flag f -vector (for general graded posets) and the cd-index (for Eulerian posets) are given in [7]. We note that in [7], this distinction between κ i and h i is not made, so their formulas for h i are, in reality, for h n−i (which equals h i in the Eulerian case).…”

“…These and more general even antichains of intervals have been related to extremes of the cone of Eulerian flag vectors in [8,9].…”

“…The main result [2, Theorem 4.1] is that the combinatorial Hopf algebra (QSym, ζ Q ) is a terminal object in the category of combinatorial Hopf algebras, that is, for any combinatorial Hopf algebra (H, ζ H ), there is a unique morphism F : (H, ζ H ) → (QSym, ζ Q ). For (H, ζ H ) = (P, ζ) from Example 6, the morphism F is the one given in (7).…”

“…Now Π QSym = Π, the peak algebra with basis given in (8), and Π P contains the subalgebra of all Eulerian posets. Together, this gives another proof of Theorem 3.5.…”