138 research outputs found

    Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

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    We prove that the noncrossing partition lattices associated with the complex reflection groups G(d,d,n)G(d,d,n) for d,n≥2d,n\geq 2 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 γ\gamma-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

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    We address a supersaturation problem in the context of forbidden subposets. A family F\mathcal{F} of sets is said to contain the poset PP if there is an injection i:P→Fi:P \rightarrow \mathcal{F} such that p≤Pqp \le_P q implies i(p)⊂i(q)i(p) \subset i (q). The poset on four elements a,b,c,da,b,c,d with a,b≤c,da,b \le c,d is called butterfly. The maximum size of a family F⊆2[n]\mathcal{F} \subseteq 2^{[n]} that does not contain a butterfly is Σ(n,2)=(n⌊n/2⌋)+(n⌊n/2⌋+1)\Sigma(n,2)=\binom{n}{\lfloor n/2 \rfloor}+\binom{n}{\lfloor n/2 \rfloor+1} as proved by De Bonis, Katona, and Swanepoel. We prove that if F⊆2[n]\mathcal{F} \subseteq 2^{[n]} contains Σ(n,2)+E\Sigma(n,2)+E sets, then it has to contain at least (1−o(1))E(⌈n/2⌉+1)(⌈n/2⌉2)(1-o(1))E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2} copies of the butterfly provided E≤2n1−εE\le 2^{n^{1-\varepsilon}} for some positive ε\varepsilon. We show by a construction that this is asymptotically tight and for small values of EE we show that the minimum number of butterflies contained in F\mathcal{F} is exactly E(⌈n/2⌉+1)(⌈n/2⌉2)E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}

    The structure of the consecutive pattern poset

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    The consecutive pattern poset is the infinite partially ordered set of all permutations where σ≤τ\sigma\le\tau if τ\tau has a subsequence of adjacent entries in the same relative order as the entries of σ\sigma. 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

    Sperner type theorems with excluded subposets

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    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

    On the Duality of Semiantichains and Unichain Coverings

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    We study a min-max relation conjectured by Saks and West: For any two posets PP and QQ the size of a maximum semiantichain and the size of a minimum unichain covering in the product P×QP\times Q are equal. For positive we state conditions on PP and QQ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure

    AZ-identities and Strict 2-part Sperner Properties of Product Posets

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    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
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