170 research outputs found
Sperner type theorems with excluded subposets
Let F be a family of subsets of an n-element set. Sperner's theorem says that if there is no inclusion among the members of F then the largest family under this condition is the one containing all ⌊ frac(n, 2) ⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of F is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let P be a poset. The maximum size of a family F which does not contain P as a (not-necessarily induced) subposet is denoted by La (n, P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11-13, 2009], but it was somewhat updated in December 2010. © 2011 Elsevier B.V. All rights reserved
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
Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued
Richard Stanley played a crucial role, through his work and his students, in
the development of the relatively new area known as combinatorial
representation theory. In the early stages, he has the merit to have pointed
out to combinatorialists the potential that representation theory has for
applications of combinatorial methods. Throughout his distinguished career, he
wrote significant articles which touch upon various combinatorial aspects
related to representation theory (of Lie algebras, the symmetric group, etc.).
I describe some of Richard's contributions involving Lie algebras, as well as
recent developments inspired by them (including some open problems), which
attest the lasting impact of his work.Comment: 11 page
AZ-identities and Strict 2-part Sperner Properties of Product Posets
One of the central issues in extremal set theory is Sperner's theorem and its
generalizations. Among such generalizations is the best-known BLYM inequality
and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM
inequality into an identity. Sperner's theorem and the BLYM inequality has been
also generalized to a wide class of posets. Another direction in this research
was the study of more part Sperner systems. In this paper we derive AZ type
identities for regular posets. We also characterize all maximum 2-part Sperner
systems for a wide class of product posets
The structure of the consecutive pattern poset
The consecutive pattern poset is the infinite partially ordered set of all
permutations where if has a subsequence of adjacent
entries in the same relative order as the entries of . We study the
structure of the intervals in this poset from topological, poset-theoretic, and
enumerative perspectives. In particular, we prove that all intervals are
rank-unimodal and strongly Sperner, and we characterize disconnected and
shellable intervals. We also show that most intervals are not shellable and
have M\"obius function equal to zero.Comment: 29 pages, 7 figures. To appear in IMR
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