D’Andrea, Alessandro; De Sanctis, Luca The Kruskal-Katona theorem and a characterization of system signatures. (English) Zbl 1360.94305 J. Appl. Probab. 52, No. 2, 508-518 (2015). Summary: We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of J. B. Kruskal [Math. Optimization Tech. 251–278 (1963; Zbl 0116.35102)] and G. Katona [in: Theory of Graphs, Proc. Colloquium Tihany, Hungary 1966, 187–207 (1968; Zbl 0313.05003)]. We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution. Cited in 10 Documents MSC: 94A60 Cryptography 60C05 Combinatorial probability Keywords:signature; reliability; simplicial complexes Citations:Zbl 0116.35102; Zbl 0313.05003 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38 , 597-603. · Zbl 1042.62090 · doi:10.1239/jap/996986765 [2] Gertsbakh, I. (2000). Reliability Theory. With Applications to Preventive Maintenance. Springer, Berlin. · Zbl 0959.62088 [3] Katona, G. (1968). A theorem of finite sets. In Theory of Graphs, Academic Press, New York, pp. 187-207. · Zbl 0313.05003 [4] Kruskal, J. B. (1963). The number of simplices in a complex, In Mathematical Optimization Techniques, University of California Press, Berkeley, pp. 251-278. · Zbl 0116.35102 [5] Navarro, J. and Rubio, R. (2010). Computations of signatures of coherent systems with five components. Comm. Statist. Simul. Comput. 39 , 68-84. · Zbl 1183.62178 · doi:10.1080/03610910903312185 [6] Ross, S. M., Shahshahani, M. and Weiss, G. (1980). On the number of component failures in systems whose component lives are exchangeable. Math. Operat. Res. 5 , 358-365. · Zbl 0442.90028 · doi:10.1287/moor.5.3.358 [7] Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability, (Internat. Ser. Operat. Res. Manag. Sci. 110 ). Springer, New York. · Zbl 1154.62075 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.