602 research outputs found
Generalizations of Eulerian partially ordered sets, flag numbers, and the Mobius function
A partially ordered set is r-thick if every nonempty open interval contains
at least r elements. This paper studies the flag vectors of graded, r-thick
posets and shows the smallest convex cone containing them is isomorphic to the
cone of flag vectors of all graded posets. It also defines a k-analogue of the
Mobius function and k-Eulerian posets, which are 2k-thick. Several
characterizations of k-Eulerian posets are given. The generalized
Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A
new inequality is proved to be valid and sharp for rank 8 Eulerian posets
Signs in the cd-index of Eulerian partially ordered sets
A graded partially ordered set is Eulerian if every interval has the same
number of elements of even rank and of odd rank. Face lattices of convex
polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector
can be encoded efficiently in the cd-index. The cd-index of a polytope has all
positive entries. An important open problem is to give the broadest natural
class of Eulerian posets having nonnegative cd-index. This paper completely
determines which entries of the cd-index are nonnegative for all Eulerian
posets. It also shows that there are no other lower or upper bounds on
cd-coefficients (except for the coefficient of c^n)
On the non-existence of an R-labeling
We present a family of Eulerian posets which does not have any R-labeling.
The result uses a structure theorem for R-labelings of the butterfly poset.Comment: 6 pages, 1 figure. To appear in the journal Orde
Level Eulerian Posets
The notion of level posets is introduced. This class of infinite posets has
the property that between every two adjacent ranks the same bipartite graph
occurs. When the adjacency matrix is indecomposable, we determine the length of
the longest interval one needs to check to verify Eulerianness. Furthermore, we
show that every level Eulerian poset associated to an indecomposable matrix has
even order. A condition for verifying shellability is introduced and is
automated using the algebra of walks. Applying the Skolem--Mahler--Lech
theorem, the -series of a level poset is shown to be a rational
generating function in the non-commutative variables and .
In the case the poset is also Eulerian, the analogous result holds for the
-series. Using coalgebraic techniques a method is developed to
recognize the -series matrix of a level Eulerian poset
The short toric polynomial
We introduce the short toric polynomial associated to a graded Eulerian
poset. This polynomial contains the same information as the two toric
polynomials introduced by Stanley, but allows different algebraic
manipulations. The intertwined recurrence defining Stanley's toric polynomials
may be replaced by a single recurrence, in which the degree of the discarded
terms is independent of the rank. A short toric variant of the formula by Bayer
and Ehrenborg, expressing the toric -vector in terms of the -index, may
be stated in a rank-independent form, and it may be shown using weighted
lattice path enumeration and the reflection principle. We use our techniques to
derive a formula expressing the toric -vector of a dual simplicial Eulerian
poset in terms of its -vector. This formula implies Gessel's formula for the
toric -vector of a cube, and may be used to prove that the nonnegativity of
the toric -vector of a simple polytope is a consequence of the Generalized
Lower Bound Theorem holding for simplicial polytopes.Comment: Minor correction
Inequalities for the h- and flag h-vectors of geometric lattices
We prove that the order complex of a geometric lattice has a convex ear
decomposition. As a consequence, if D(L) is the order complex of a rank (r+1)
geometric lattice L, then for all i \leq r/2 the h-vector of D(L) satisfies
h(i-1) \leq h(i) and h(i) \leq h(r-i).
We also obtain several inequalities for the flag h-vector of D(L) by
analyzing the weak Bruhat order of the symmetric group. As an application, we
obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of
geometric lattices which minimize Whitney numbers of the second kind. In
addition, we are able to give a combinatorial flag h-vector proof of h(i-1)
\leq h(i) when i \leq (2/7)(r + 5/2).Comment: 15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was
added regarding a solution to problem 4.
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