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Two-terminal series-parallel networks

Published online by Cambridge University Press:  01 July 2016

Z. A. Lomnicki
Affiliation:
Codsall, Staffs

Abstract

This paper discusses two-terminal series-parallel networks occurring in Applied Probability and other contexts where the mathematical theory of reliability is developed. Eighty years ago Mac Mahon successfully discussed the number of different structures built of n identical components [1]; later Knödel [3] and Carlitz and Riordan [4] in the 1950's investigated the number of different structures built of n different components. However, in many new applications, engineers and statisticians have to study structures having n components with a specification defining various types of components and stating how many identical components of each kind should be used. This general problem is investigated in Section 7 whilst in Sections 1–6 results obtained earlier are presented and it is shown there how modern approaches to combinatorics (particularly Pólya's Enumerative Combinatorial Analysis) can simplify some reasoning of previous authors.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Mac Mahon, P. A. (1892) The combination of resistances. The Electrician 28, 601602.Google Scholar
[2] Riordan, J. and Shannon, C. E. (1942) The number of two-terminal series-parallel networks. J. Math. and Phys. 21, 8392.CrossRefGoogle Scholar
[3] Knödel, W. (1950) Über Zerfällungen. Monatsh. Math. LV, 2027.Google Scholar
[4] Carlitz, L. and Riordan, J. (1956) The number of labeled two-terminal series-parallel networks. Duke Math J. 23, 435446.CrossRefGoogle Scholar
[5] Curry, H. B. (1963) Foundations of Mathematical Logic. McGraw Hill Book Co., New York.Google Scholar
[6] Birnbaum, Z. W., Esary, J. D. and Saunders, S. C. (1961) Multicomponent systems and structures and their reliability. Technometrics V, 5577.CrossRefGoogle Scholar
[7] Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. J. Wiley, New York.Google Scholar
[8] Riordan, J. (1958) An Introduction to Combinatorial Analysis. J. Wiley, New York.Google Scholar
[9] Pólya, G. (1937) Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und Chemische Verbindungen. Acta Math. 68, 145254.CrossRefGoogle Scholar
[10] Pólya, G. (1956) On picture writing. Amer. Math. Monthly 63, 689697.CrossRefGoogle Scholar
[11] De Bruijn, N. G. (1964) Pólya's theory of counting. Chapter 5 in Applied Combinatorial Mathematics. Ed. Beckenbach, E. F.. J. Wiley, New York.Google Scholar
[12] Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. 1. J. Wiley, New York.Google Scholar

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