Note on Generating All Subsets of a Finite Set with Disjoint Unions
نویسندگان
چکیده
منابع مشابه
Generating all subsets of a finite set with disjoint unions
If X is an n-element set, we call a family G ⊂ PX a k-generator for X if every x ⊂ X can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Sebő [10] conjectured that for n > 2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjectu...
متن کاملNote on Generating All Subsets of a Finite Set with Disjoint Unions
We call a family G ⊂ P[n] a k-generator of P[n] if every x ⊂ [n] can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Sebő [1] conjectured that for any n ≥ k, such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize a...
متن کامل2 1 N ov 2 00 8 Note on generating all subsets of a finite set with disjoint unions
We call a family G ⊂ P[n] a k-generator of P[n] if every x ⊂ [n] can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Seb˝ o [1] conjectured that for any n ≥ k, such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize...
متن کامل1 8 N ov 2 00 8 Note on generating all subsets of a finite set with disjoint unions
We call a family G ⊂ P[n] a k-generator of P[n] if every x ⊂ [n] can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Seb˝ o [1] conjectured that for any n ≥ k, such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize...
متن کاملGenerating All Sets With Bounded Unions
We consider the problem of minimizing the size of a family of sets G such that every subset of {1, . . . , n} can be written as a disjoint union of at most k members of G, where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the question of finding the minimum of |G| so that every subset of {1, ....
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2009
ISSN: 1077-8926
DOI: 10.37236/254