895 research outputs found
Linear inequalities for flags in graded posets
The closure of the convex cone generated by all flag -vectors of graded
posets is shown to be polyhedral. In particular, we give the facet inequalities
to the polar cone of all nonnegative chain-enumeration functionals on this
class of posets. These are in one-to-one correspondence with antichains of
intervals on the set of ranks and thus are counted by Catalan numbers.
Furthermore, we prove that the convolution operation introduced by Kalai
assigns extreme rays to pairs of extreme rays in most cases. We describe the
strongest possible inequalities for graded posets of rank at most 5
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)
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.
The Tchebyshev transforms of the first and second kind
We give an in-depth study of the Tchebyshev transforms of the first and
second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform
(of the first kind) preserves desirable combinatorial properties, including
Eulerianess (due to Hetyei) and EL-shellability. It is also a linear
transformation on flag vectors. When restricted to Eulerian posets, it
corresponds to the Billera, Ehrenborg and Readdy omega map of oriented
matroids. One consequence is that nonnegativity of the cd-index is maintained.
The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on
the space of quasisymmetric functions QSym. It coincides with Stembridge's peak
enumerator for Eulerian posets, but differs for general posets. The complete
spectrum is determined, generalizing work of Billera, Hsiao and van
Willigenburg.
The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's
classical quasisymmetric function of a poset, this map is a comodule morphism
with respect to the quasisymmetric functions QSym.
Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform,
and the Tchebyshev transforms motivate a general study of chain maps. One such
occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on
the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and
Sottile's result on the terminal object in the category of combinatorial Hopf
algebras. In contrast, the chain map of the first kind is both an algebra map
and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page
- …