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Permanents of Random Doubly Stochastic Matrices

Published online by Cambridge University Press:  20 November 2018

R. C. Griffiths
R. C. Griffiths*
Affiliation:
Macquarie University, N.S.W., Australia; Monash University, Clayton, 3168, Australia
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The permanent of an n × n matrix A = (aij) is defined as

where Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 600 - 607
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Marcus, M., Inequalities for matrix functions of combinatorial interest, SIAM J. Appl. Math. 17 (1969), 1023–1031.Google Scholar
2. Marcus, M. and Minc, H., Permanents, Amer. Math. Monthly 72 (1965), 577–591.Google Scholar
3. Ryser, H. R., Combinatorial mathematics, No. 14 of the Carus Mathematical Monographs, the Mathematical Association of America, 1963.Google Scholar