936 research outputs found
Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
We prove that the noncrossing partition lattices associated with the complex
reflection groups for admit symmetric decompositions
into Boolean subposets. As a result, these lattices have the strong Sperner
property and their rank-generating polynomials are symmetric, unimodal, and
-nonnegative. We use computer computations to complete the proof that
every noncrossing partition lattice associated with a well-generated complex
reflection group is strongly Sperner, thus answering affirmatively a question
raised by D. Armstrong.Comment: 30 pages, 5 figures, 1 table. Final version. The results of the
initial version were extended to symmetric Boolean decompositions of
noncrossing partition lattice
Supersaturation and stability for forbidden subposet problems
We address a supersaturation problem in the context of forbidden subposets. A
family of sets is said to contain the poset if there is an
injection such that implies . The poset on four elements with is
called butterfly. The maximum size of a family
that does not contain a butterfly is as proved by De Bonis, Katona, and
Swanepoel. We prove that if contains
sets, then it has to contain at least copies of the butterfly provided for some positive . We show by a
construction that this is asymptotically tight and for small values of we
show that the minimum number of butterflies contained in is
exactly
On ideals with the Rees property
A homogeneous ideal of a polynomial ring is said to have the Rees
property if, for any homogeneous ideal which contains , the
number of generators of is smaller than or equal to that of . A
homogeneous ideal is said to be -full if for some , where is the graded maximal
ideal of . It was proved by one of the authors that -full
ideals have the Rees property and that the converse holds in a polynomial ring
with two variables. In this note, we give examples of ideals which have the
Rees property but are not -full in a polynomial ring with more
than two variables. To prove this result, we also show that every Artinian
monomial almost complete intersection in three variables has the Sperner
property.Comment: 8 page
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper
The strong Lefschetz property for Artinian algebras with non-standard grading
We define the strong Lefschetz property for finite graded modules over graded
Artinian algebras whose grading is not necessarily standard. We show that most
results which have been obtained for Artinian algebras with standard grading
can be extended for non-standard grading. Our results on the strong Lefschetz
property for non-standard grading can be used to prove that certain Artinian
complete intersections with standard grading have the strong Lefschetz
property.Comment: 24 pages, To appear in Journal of Algebr
KKM type theorems with boundary conditions
We consider generalizations of Gale's colored KKM lemma and Shapley's KKMS
theorem. It is shown that spaces and covers can be much more general and the
boundary KKM rules can be substituted by more weaker boundary assumptions.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1406.6672 by other author
Zero-error capacity of binary channels with memory
We begin a systematic study of the problem of the zero--error capacity of
noisy binary channels with memory and solve some of the non--trivial cases.Comment: 10 pages. This paper is the revised version of our previous paper
having the same title, published on ArXiV on February 3, 2014. We complete
Theorem 2 of the previous version by showing here that our previous
construction is asymptotically optimal. This proves that the isometric
triangles yield different capacities. The new manuscript differs from the old
one by the addition of one more pag
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