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On the Permanent of a Doubly Stochastic Matrix

Published online by Cambridge University Press:  20 November 2018

Herbert S. Wilf
Herbert S. Wilf*
Affiliation:
University of Pennsylvania
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If is an n X n matrix, the permanent of A, Per A, is defined by

1

where the sum is over all permutations. If A is doubly stochastic (i.e., nonnegative with row and column sums all equal to 1), then it has been conjectured that Per A ⩾ n!/nn. When confronted with a vaguely similar problem about determinants, M. Kac (1) observed that the computation of minima can often be aided by knowledge of various averages. In this spirit we calculate here the average permanent of a class of doubly stochastic matrices and thereby obtain upper bounds for the minima. These turn out to be surprisingly sharp.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 758 - 761
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Kac, M., Probability and related topics in physical sciences, Seminar at Boulder, Colorado, Summer, 1957 (lecture notes).Google Scholar
2. Nikolai, P. J., Permanents of incidence matrices, MTAC(1960), 262–266.Google Scholar
3. Ryser, H. J., Matrices of zeros and ones, Bull. Amer. Math. Soc, 66 (1960), 442–464.Google Scholar