论文标题
有限集的建设性封面
Constructive covers of a finite set
论文作者
论文摘要
给定积极的整数$ n，k $带有$ k \ leq n $，我们考虑选择$ k $ subset的$ \ {1，\ ldots，n \} $的方法，以使这些子集的结合给出$ \ {1，\ ldots，n \} $，并且它们不是彼此的子集。我们将这种选择的选择称为建设性$ k $ - 覆盖物，并提供一个半分析的求和公式来计算$ \ {1，\ ldots，n \} $的确切构建性$ k $ - covers。总结中的每个术语都是第二类Stirl数字的新变体的乘积，称为集成的Stirling号码，以及某个集合的基数，我们通过基于优化的过程计算出，该过程对二进制变量进行了无良好的剪辑。
Given positive integers $n,k$ with $k\leq n$, we consider the number of ways of choosing $k$ subsets of $\{1,\ldots,n\}$ in such a way that the union of these subsets gives $\{1,\ldots,n\}$ and they are not subsets of each other. We refer to such choices of sets as constructive $k$-covers and provide a semi-analytic summation formula to calculate the exact number of constructive $k$-covers of $\{1,\ldots,n\}$. Each term in the summation is the product of a new variant of Stirling numbers of the second kind, referred to as integrated Stirling numbers, and the cardinality of a certain set which we calculate by an optimization-based procedure with no-good cuts for binary variables.
