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The asymptotic behaviour of the number of t -minimal coverings of a set

The asymptotic behaviour of the number of t -minimal coverings of a set -- The concept of a £-minimal covering of a set is introduced; exact and asymptotic formulae for the number of such coverings are obtained. 1. INTRODUCTION Let X = {zi,..., xn}, and let X\,..., Xk be a family of different non-empty subsets of the set X\ these ,..., Xk are called blocks. A family ,..., Xk is called a i-minimal covering of an n-set X, if X - X\ U ... U A'* and for any -tuple \<i\<...<it<k there exists an element £ X such that * e *,·,,...,* G ,·,; ^ ,...,^ - , (1) where {tV..,t t } U {ji,...,jk-t} = {!,...,*}, {*i,.·.,*«} {ji,...,j k -t} = 0. We say that the element is -fold covered. It is clear that for different £-tuples the corresponding elements of the set X are distinct. For t = 1 the notion of *-minimal covering coincides with the notion of minimal covering, which was investigated in [1-3]. With any 2-minimal covering of an -set by k blocks we associate the (0, l)-matrix A, which is the incidence matrix of the family of blocks. For a £-tuple I < i\ < ... < it < k the column of the matrix http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

The asymptotic behaviour of the number of t -minimal coverings of a set

SACHKOV, V. N.
Discrete Mathematics and Applications , Volume 3 (3) – Jan 1, 1993
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References (3)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.3.265
Publisher site
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Abstract

-- The concept of a £-minimal covering of a set is introduced; exact and asymptotic formulae for the number of such coverings are obtained. 1. INTRODUCTION Let X = {zi,..., xn}, and let X\,..., Xk be a family of different non-empty subsets of the set X\ these ,..., Xk are called blocks. A family ,..., Xk is called a i-minimal covering of an n-set X, if X - X\ U ... U A'* and for any -tuple \<i\<...<it<k there exists an element £ X such that * e *,·,,...,* G ,·,; ^ ,...,^ - , (1) where {tV..,t t } U {ji,...,jk-t} = {!,...,*}, {*i,.·.,*«} {ji,...,j k -t} = 0. We say that the element is -fold covered. It is clear that for different £-tuples the corresponding elements of the set X are distinct. For t = 1 the notion of *-minimal covering coincides with the notion of minimal covering, which was investigated in [1-3]. With any 2-minimal covering of an -set by k blocks we associate the (0, l)-matrix A, which is the incidence matrix of the family of blocks. For a £-tuple I < i\ < ... < it < k the column of the matrix

Journal

Discrete Mathematics and Applications – de Gruyter

Published: Jan 1, 1993

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