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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An operator approach to the principle of inclusion and exclusion


Author: C. J. Liu
Journal: Proc. Amer. Math. Soc. 96 (1986), 528-536
MSC: Primary 05A15; Secondary 05C30
MathSciNet review: 822454
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Abstract: Using an operator approach we derive Sylvester-Whitworth formulae for sets $ A$'s. By the same token we treat the problem where both sets of $ A$'s and $ B$'s are involved. Our result extends the Sylvester-Whitworth inclusion and exclusion formula to the resolution of the number of elements in exactly $ {m_1}$ sets of $ A$'s and $ {m_2}$ sets of $ B$'s respectively. The formula are applied to the complete graph and complete bipartite graph. The enumeration of spanning subgraphs with any preassigned number of disconnected cycles is solved, together with the case where any preassigned number of vertices have degree one.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0822454-1
PII: S 0002-9939(1986)0822454-1
Article copyright: © Copyright 1986 American Mathematical Society