On the monotonicity of the permanent

Author:
Suk Geun Hwang

Journal:
Proc. Amer. Math. Soc. **106** (1989), 59-63

MSC:
Primary 15A15; Secondary 15A51

DOI:
https://doi.org/10.1090/S0002-9939-1989-0960645-2

MathSciNet review:
960645

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the set of all doubly stochastic matrices and let . For , if is a nondecreasing function of on , we say that the monotonicity of permanent (abb. MP) holds for . Friedland and Mine [3] proved MP for . In [6], Lih and Wang proposed a problem of determining whether MP holds for .

In this note, we prove MP for , extending the result of Friedland and Mine, and give an affirmative answer to the Lih and Wang's question.

**[1]**R. Bapat,*Doubly stochastic matrices with equal subpermanents*, Linear Algebra Appl.**51**(1983), 1-8. MR**699719 (84g:15024)****[2]**G. P. Egorycev,*A solution of van der Waerden's permanent problem*, Kirenski Institute of Phisics, Acad. of Sci. SSSR, Preprint IFSO-13M, Krasnoyarsk 1980 (Russian). MR**602332 (82e:15006)****[3]**S. Friedland and H. Minc,*Monotonicity of permanents of doubly stochastic matrices*, Linear and Multilinear Algebra**6**(1978), 227-231. MR**512998 (80b:15013)****[4]**J. Goldwasser,*Contributions to the theory of permanents*, Ph.D. Thesis, Univ. of Wisconsin-Madison, 1983.**[5]**S. G. Hwang,*The monotonicity and theokvić conjectures on permanents of doubly stochastic matrices*, Linear Algebra Appl.**79**(1986), 127-151. MR**847194 (87j:15017)****[6]**K.-W. Lih and E. T. H. Wang,*Monotonicity conjecture on permanents of doubly stochastic matrices*, Proc. Amer. Math. Soc.**82**(1981), 173-178. MR**609645 (82d:15003)****[7]**M. Marcus and R. Merris,*A relation between permanental and determinantal adjoints*, J. Austral. Math. Soc.**15**(1973), 270-271. MR**0335533 (49:314)****[8]**H. Minc,*Theory of permanents 1978-1981*, Linear and Multilinear Algebra**12**(1983), 227-263. MR**688551 (84g:15008)****[9]**-,*Permanents, Encyclopedia of Math. and Its Appl*., Addison-Wesley, Reading, Mass., 1978. MR**504978 (80d:15009)****[10]**R. Sinkhorn,*Doubly stochastic matrices with dominant**-minors*, Linear and Multilinear Algebra**5**(1977), 107-117. MR**0444688 (56:3038)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0960645-2

Keywords:
Permanent,
doubly stochastic matrix,
monotonicity

Article copyright:
© Copyright 1989
American Mathematical Society