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Proceedings of the American Mathematical Society

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Conversion of the permanent into the determinant


Author: P. M. Gibson
Journal: Proc. Amer. Math. Soc. 27 (1971), 471-476
MSC: Primary 15.20
DOI: https://doi.org/10.1090/S0002-9939-1971-0279110-X
MathSciNet review: 0279110
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be an $ n$-square $ (0,1)$-matrix with positive permanent. It is shown that if the permanent of $ A$ can be converted into a determinant by affixing $ \pm $ signs to the elements of $ A$ then $ A$ has at most $ ({n^2} + 3n - 2)/2$ positive entries. Corollaries of this result are given.


References [Enhancements On Off] (What's this?)

  • [1] P. M. Gibson, An identity between permanents and determinants, Amer. Math. Monthly 76 (1969), 270-271. MR 39 #2779. MR 0241439 (39:2779)
  • [2] M. Marcus and H. Minc, On the relation between the determinant and the permanent, Illinois J. Math. 5 (1961), 376-381. MR 26 #5004. MR 0147488 (26:5004)
  • [3] H. Minc, On lower bounds for permanents of $ (0,1)$ matrices, Proc. Amer. Math Soc. 22 (1969), 117-123. MR 39 #6891. MR 0245585 (39:6891)
  • [4] G. Pólya, Aufgabe 424, Arch. Math. Phys. (3) 20 (1913), 271.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0279110-X
Keywords: Permanents, determinants, $ (0,1)$-matrices, matrices over a field
Article copyright: © Copyright 1971 American Mathematical Society

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