142 research outputs found
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)
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
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
Cyclotomic factors of the descent set polynomial
We introduce the notion of the descent set polynomial as an alternative way
of encoding the sizes of descent classes of permutations. Descent set
polynomials exhibit interesting factorization patterns. We explore the question
of when particular cyclotomic factors divide these polynomials. As an instance
we deduce that the proportion of odd entries in the descent set statistics in
the symmetric group S_n only depends on the number on 1's in the binary
expansion of n. We observe similar properties for the signed descent set
statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up
notatio
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