Problems on chain partitions
TLDR
This chapter discusses problems about the poset of subsets of a finite set, and gives problems for other families of posets.About:
This article is published in Discrete Mathematics.The article was published on 1988-12-01 and is currently open access. It has received 32 citations till now. The article focuses on the topics: Partially ordered set & Finite set.read more
Citations
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Journal ArticleDOI
Graph colorings and related symmetric functions: ideas and applications 1 A description of results, interesting applications, & notable open problems
TL;DR: The question of when the expansion of X G in terms of Schur functions has nonnegative coefficients is considered and a number of applications are given, including new conditions on the f -vector of a flag complex and a new class of polynomials with real zeros.
A symmetric chain decomposition of L(4,n)
TL;DR: It is verified that L(m, n), the set of integer m-tuples, is a symmetric chain order for all ( m, n) by construction for m = 4.
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Partitioning the Boolean Lattice into Chains of Large Minimum Size
TL;DR: It is shown that for 0?j?n, the union of the lowest j+1 elements from each of the chains in the CSCD of 2n] forms a poset Tj(n) with the normalized matching property and log-concave rank numbers, which is used to show that the nodes in theCSCD chains of size less than 2d( n) may be repartitioned into chains of large minimum size, as desired.
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Proof of a conjecture on partitions of a Boolean lattice
TL;DR: In this paper, it was shown that if n is sufficiently large given c then the Boolean lattice consisting of all subsets of an n-element set can be partitioned into chains of size c except for at most c − 1 elements which also form a chain.
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Partitioning the boolean lattice into a minimal number of chains of relatively uniform size
TL;DR: It is shown that for any c > 1, there exist functions e(n) ∼ √n/2 and f( n)∼ c√n log n and an integer N (depending only on c) such that for all n < N, there is a chain decomposition of the Boolean lattice 2[n] into (n ⌊n/ 2⌋) chains.
References
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Book ChapterDOI
A decomposition theorem for partially ordered sets
TL;DR: In this article, a partially ordered set P is considered and two elements a and b of P are camparable if either a ≧ b or b ≧ a. If b and a are non-comparable, then they are independent.
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Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property
TL;DR: Techniques from algebraic geometry are used to show that certain finite partially ordered sets Q^X derived from a class of algebraic varieties X have the k-Sperner property for all k, which means that there is a simple description of the cardinality of the largest subset of $Q^X $ containing no $( k + 1 )$-element chain.
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The structure of sperner k-families
Curtis Greene,Daniel J. Kleitman +1 more
TL;DR: An extension of Dilworth's theorem is obtained by relating the maximum size of a k-family to certain partitions of P into chains, and a natural lattice ordering on k-families is defined and analyzed.