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An expansion for the permanent of a doubly stochastic matrix

Published online by Cambridge University Press:  09 April 2009

R. C. Griffiths
R. C. Griffiths
Affiliation:
Macquarie UniversityNorth Ryde, N. S. W. 2113 Australia
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The permanent of an n-square matrix A = (aij) is defined by where Sn is the symmetric group of order n. Kn will denote the convex set of all n-square doubly stochastic matrices and K0n its interior. Jn ∈ Kn will be the matrix with all elements equal to 1/n. If M ∈ K0n, then M lies on a line segment passing through Jn and another B ∈ Kn — K0n. This note gives an expansion for the permanent of such a line segment as a weighted average of permanents of matrices in Kn. For a survey article on permanents the reader is referred to Marcus and Mine [3].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 4 , June 1973 , pp. 504 - 509
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Marcus, M. and Newman, M., ‘On the minimum of the permanent of a doubly stochastic matrix’, Duke Math. Jour. 26 (1959), 61–72.Google Scholar
[2]Marcus, M. and Newman, M., ‘Inequalities for the permanent function’, Ann. Math. 75 (1962), 47–62.Google Scholar
[3]Marcus, M. and Minc, H., ‘Permanents’, Amer. Math. Monthly 72 (1965), 577–591.Google Scholar
[4]van der Waerden, B. L., ‘Aufgabe 45’, Jber. Deutsch. Math.-Verein. 35 (1926), 117.Google Scholar