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Bonferroni-Galambos inequalities for partition lattices. (English) Zbl 1062.05015

Summary: We establish a new analogue of the classical Bonferroni inequalities and their improvements by Galambos for sums of type \(\sum_{\pi\in {\mathbb P}(U)} (-1)^{|\pi|-1} (|\pi|-1)! f(\pi)\) where \(U\) is a finite set, \({\mathbb P}(U)\) is the partition lattice of \(U\) and \(f:{\mathbb P}(U)\rightarrow{\mathbb R}\) is some suitable non-negative function. Applications of this new analogue are given to counting connected \(k\)-uniform hypergraphs, network reliability, and cumulants.

MSC:

05A18 Partitions of sets
60C05 Combinatorial probability
60E15 Inequalities; stochastic orderings
05C65 Hypergraphs
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