An update on Minc's survey of open problems involving permanents

Linear Algebra and Its Applications

Volumn 403, Issue 1-3, 2005, Pages 314-342

  Cheon, Gi Sang ^a   Wanless, Ian M ^b  

^a Daejin University   (South Korea)

^b CHARLES DARWIN UNIVERSITY   (Australia)

Author keywords

Circulant; Doubly stochastic; Permanent; Permanental dominance conjecture; Positive semidefinite Hermitian; Subpermanent sum

Indexed keywords

ALGORITHMS; COMBINATORIAL MATHEMATICS; MATRIX ALGEBRA; RANDOM PROCESSES; SURVEYS; THEOREM PROVING;

CIRCULANT; DOUBLY STOCHASTIC; PERMANENT; PERMANENTAL DOMINANCE CONJECTURE; POSITIVE SEMIDEFINITE HERMITIAN; SUBPERMANENT SUM;

PROBLEM SOLVING;

EID: 18844367229     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2005.02.030     Document Type: Article

References (106)

1. R.B. Bapat, and V.S. Sunder On majorization and Schur products Linear Algebra Appl. 72 1985 107 117
2. R.B. Bapat, and V.S. Sunder An extremal property of the permanent and the determinant Linear Algebra Appl. 76 1986 153 163
3. L.B. Beasley An inequality on permanents of Hadamard products Bull. Korean Math. Soc. 37 2000 633 639
4. A. Bernasconi, B. Codenotti, V. Crespi, and G. Resta How fast can one compute the permanent of circulant matrices? Linear Algebra Appl. 292 1999 15 37
5. V.I. Bol'shakov The spectrum of the permanent on Λ n k (Russian) O.B. Lupanov Proceedings of the All-Union seminar on discrete mathematics and its applications 1986 Moskov. Gos. Univ. Moscow 65 73
6. L.M. Brègman Some properties of nonnegative matrices and their permanents Soviet Math. Dokl. 14 1973 945 949
7. R.A. Brualdi An interesting face of the polytope of doubly stochastic matrices Linear and Multilinear Algebra 17 1985 5 18
8. R.A. Brualdi, J.L. Goldwasser, and T.S. Michael Maximum permanents of matrices of zeroes and ones J. Combin. Theory Ser. A 47 1988 207 245
9. O. Chan, and T.K. Lam Hook immanantal inequalities for Laplacians of trees Linear Algebra Appl. 261 1997 23 47
10. D.K. Chang Minimum and maximum permanents of certain doubly stochastic matrices Linear and Multilinear Algebra 24 1988 39 44
11. D.K. Chang On two permanental conjectures Linear and Multilinear Algebra 26 1990 207 213
12. G.-S. Cheon, and S.-G. Hwang Maximization of a matrix function related to the Dittert conjecture Linear Algebra Appl. 165 1992 153 165
13. J. Chollet Permanents and Hadamard products Amer. Math. Monthly 89 1982 57 58
14. B. Codenotti, and G. Resta Computation of sparse circulant permanents via determinants Linear Algebra Appl. 355 2002 15 34
15. P. Dagum, and M. Luby Approximating the permanent of graphs with large factors Theoret. Comput. Sci. 102 1992 283 305
16. n having irrational minimum permanents Linear and Multilinear Algebra 30 1991 145 154
17. D.Ž. Djokovic On a conjecture by van der Waerden Mat. Vesnik 19 1967 566 569
18. J.H. Drew, and C.R. Johnson The maximum permanent of a 3-by-3 positive semidefinite matrix, given the eigenvalues Linear and Multilinear Algebra 25 1989 243 251
19. J.H. Drew, and C.R. Johnson Counterexample to a conjecture of Mehta regarding permanental maximization Linear and Multilinear Algebra 25 1989 253 254
20. I. Fischer, and S.-G. Hwang Certain nonbarycentric cohesive matrices Linear Algebra Appl. 239 1996 185 200
21. P. Flor Research problem Bull. Amer. Math. Soc. 72 1966 30
22. T.H. Foregger On the minimum value of the permanent of a nearly decomposable doubly stochastic matrix Linear Algebra Appl. 32 1980 75 85
23. T.H. Foregger Minimum permanents of multiplexes Linear Algebra Appl. 87 1987 197 211
24. T.H. Foregger Permanents of convex combinations of doubly stochastic matrices Linear and Multilinear Algebra 23 1988 79 90
25. T.H. Foregger, and R. Sinkhorn On matrices minimizing the permanent on faces of the polyhedron of the doubly stochastic matrices Linear and Multilinear Algebra 19 1986 395 397
26. S. Friedland, and H. Minc Monotonicity of permanents of doubly stochastic matrices Linear and Multilinear Algebra 6 1978/79 227 231
27. C.D. Godsil, and B.D. McKay Asymptotic enumeration of Latin rectangles J. Combin. Theory Ser. B 48 1990 19 44
28. J. Goldwasser Monotonicity of permanents of direct sums of doubly stochastic matrices Linear and Multilinear Algebra 33 1993 185 188
29. R.J. Gregorac, and I.R. Hentzel A note on the analogue of Oppenheim's inequality for permanents Linear Algebra Appl. 94 1987 109 112
30. R. Grone, C.R. Johnson, E. de Sá, and H. Wolkowicz A note on maximizing the permanent of a positive definite Hermitian matrix, given the eigenvalues Linear and Multilinear Algebra 19 1986 389 393
31. R. Grone, and R. Merris Conjectures on permanents Linear and Multilinear Algebra 21 1987 419 427
32. R. Grone, and S. Pierce Permanental inequalities for correlation matrices SIAM J. Matrix Anal. Appl. 9 1988 194 201
33. B. Gyires On permanent inequalities Colloq. Math. Soc. János Bolyai 18 1978 471 484
34. J.R. Henderson Permanents of (0, 1)-matrices having at most two 0's per line Canad. Math. Bull. 18 1975 353 358
35. P. Heyfron Immanant dominance orderings for hook partitions Linear and Multilinear Algebra 24 1988 65 78
36. F. Holens, Two aspects of doubly stochastic matrices: Permutation matrices and the minimum permanent function, Ph.D. Thesis, University of Manitoba, 1964
37. S.-M. Hong, S.-J. Kim, Y.-B. Jun, and S.-Z. Song A cohesive matrix in a conjecture on permanents Bull. of Korean Math. Soc. 33 1996 127 133
38. S.-G. Hwang On a conjecture of E. Dittert Linear Algebra Appl. 95 1987 161 169
39. S.-G. Hwang Some permanental inequalities Bull. Korean Math. Soc. 26 1989 35 42
40. S.-G. Hwang On the monotonicity of the permanent Proc. Amer. Math. Soc. 106 1989 59 63
41. G. James Permanents, immanants, and determinants Proc. Sympos. Pure Math. Part 2 47 1987 431 436
42. G. James Immanants Linear and Multilinear Algebra 32 1992 197 210
43. G. James, and M. Liebeck Permanents and immanants of Hermitian matrices Proc. London Math. Soc. 55 3 1987 243 265
44. Z. Jiao On a conjecture of H. Minc Linear and Multilinear Algebra 32 1992 103 105
45. C.R. Johnson, and S. Pierce Permanental dominance of the normalized single-hook immanants on the positive semidefinite matrices Linear and Multilinear Algebra 21 1987 215 229
46. C.R. Johnson, and S. Pierce Inequalities for single-hook immanants Linear Algebra Appl. 102 1988 55 79
47. C.S. Karuppan Chetty, and S. Maria Arulraj Falsity of Wang's conjecture on stars Linear Algebra Appl. 277 1998 49 56
48. K.H. Kim, and F.W. Roush Expressions for certain minors and permanents Linear Algebra Appl. 41 1981 93 97
49. K.A. Kopotun On some permanental conjectures Linear and Multilinear Algebra 36 1994 205 216
50. K.A. Kopotun A note on the convexity of the sum of subpermanents Linear Algebra Appl. 245 1996 157 169
51. A.R. Kräuter, Recent results on permanents of (1, -1) matrices, Ber. No. 249, Berichte, 243-254, Forschungszentrum Graz, Graz, 1985
52. A.R. Kräuter, and N. Seifter On some questions concerning permanents of (1, -1)-matrices Israel J. Math. 45 1983 53 62
53. H. Liang, and F. Bai An upper bound for the permanent of (0, 1)-matrices Linear Algebra Appl. 377 2004 291 295
54. E.H. Lieb Proofs of some conjectures on permanents J. Math. Mech. 16 1966 127 134
55. K.W. Lih, and E.T.H. Wang A convexity inequality on the permanent of doubly stochastic matrices Congr. Numer. 36 1982 189 198
56. M. Malek A note on a permanental conjecture of M. Marcus and H. Minc Linear and Multilinear Algebra 25 1989 71 73
57. M. Malek On the convex combinations of matrices and the sum of subpermanents Linear and Multilinear Algebra 26 1990 293 297
58. M. Marcus, and H. Minc Permanents Amer. Math. Monthly 72 1965 577 591
59. M. Marcus, and H. Minc On a conjecture of B.L. van der Waerden Proc. Cambridge Philos. Soc. 63 1967 305 309
60. M. Marcus, and M. Sandy Bessel's inequality in tensor space Linear and Multilinear Algebra 23 1988 233 249
61. B.D. McKay, and I.M. Wanless Maximising the permanent of (0, 1)-matrices and the number of extensions of Latin rectangles Electron. J. Combin. 5 1998 R11
62. M.L. Mehta Elements of matrix theory 1977 Hindustan Publishing Corp. Delhi
63. D. Merriell The maximum permanent in Λ n k Linear and Multilinear Algebra 9 1980 81 91
64. R. Merris The permanent of a doubly stochastic matrix Amer. Math. Monthly 80 1973 791 793
65. R. Merris Single-hook characters and Hamiltonian circuits Linear and Multilinear Algebra 14 1983 21 35
66. R. Merris The permanental dominance conjecture Current trends in matrix theory 1987 North-Holland New York 213 223
67. R. Merris Applications of multilinear algebra Linear and Multilinear Algebra 32 1992 211 224
68. H. Minc Permanents Encyclopedia Math. Appl. 6 1978 Addison-Wesley Reading, MA
69. H. Minc Theory of permanents 1978-1981 Linear and Multilinear Algebra 12 1983 227 263
70. H. Minc Theory of permanents 1982-1985 Linear and Multilinear Algebra 21 1987 109 148
71. H. Minc Research problems: permanental compounds Linear and Multilinear Algebra 19 1986 199 201
72. E. Nemeth, J. Seberry, and M. Shu On the distribution of the permanent of cyclic (0, 1)-matrices Utilitas Math. 16 1979 171 182
73. T.H. Pate Permanental dominance and the Soules conjecture for certain right ideals in the group algebra Linear and Multilinear Algebra 24 1989 135 149
74. T.H. Pate Partitions, irreducible characters, and inequalities for generalized matrix functions Trans. Amer. Math. Soc. 325 1991 875 894
75. T.H. Pate Descending chains of immanants Linear Algebra Appl. 162-4 1992 639 650
76. T.H. Pate Immanant inequalities and partition node diagrams J. London Math. Soc. 46 2 1992 65 80
77. T.H. Pate Immanant inequalities, induced characters, and rank two partitions J. London Math. Soc. 49 2 1994 40 60
78. T.H. Pate Inequalities involving immanants Linear Algebra Appl. 212/213 1994 31 44
79. T.H. Pate A machine for producing inequalities involving immanants and other generalized matrix functions Linear Algebra Appl. 254 1997 427 466
80. T.H. Pate Row appending maps, Ψ-functions, and immanant inequalities for Hermitian positive semi-definite matrices Proc. London Math. Soc. 76 3 1998 307 358
81. T.H. Pate Tensor inequalities, ξ-functions and inequalities involving immanants Linear Algebra Appl. 295 1999 31 59
82. G. Resta, and G. Sburlati On the number of different permanents of some sparse (0, 1)-circulant matrices Linear Algebra Appl. 375 2003 197 209
83. A. Schrijver Counting 1-factors in regular bipartite graphs J. Combin. Theory Ser. B 72 1998 122 135
84. A. Schrijver, and W.G. Valiant On lower bounds for permanents Nederl. Akad. Wetensch. Indag. Math. 42 1980 425 427
85. I. Schur Über endliche Gruppen und Hermitische Formen Math. Z. 1 1918 184 207
86. R. Sinkhorn A neighborhood in which the van der Waerden permanent conjecture is valid Linear and Multilinear Algebra 10 1981 217 221
87. S.-Z. Song Minimum permanents on certain faces of matrices containing an identity submatrix Linear Algebra Appl. 108 1988 263 280
88. S.-Z. Song A conjecture on permanents Linear Algebra Appl. 222 1995 91 95
89. S.-Z. Song, S.-M. Hong, Y.-B. Jun, H.-K. Kim, and S.-J. Kim On the minimum permanents related with certain barycentric matrices J. Korean Math. Soc. 34 1997 825 839
90. G.W. Soules An approach to the permanental-dominance conjecture Linear Algebra Appl. 201 1994 211 229
91. G.W. Soules Extending the Minc-Brègman upper bound for the permanent Linear and Multilinear Algebra 47 2000 77 91
92. G.W. Soules New permanental upper bounds for nonnegative matrices Linear and Multilinear Algebra 51 2003 319 337
93. G.W. Soules Permanental bounds for nonnegative matrices via decomposition Linear Algebra Appl. 394 2005 73 89
94. J.R. Stembridge Immanants of totally positive matrices are nonnegative Bull. London Math. Soc. 23 1991 422 428
95. J.R. Stembridge Some conjectures for immanants Canad. J. Math. 44 1992 1079 1099
96. E.T.H. Wang On permanents of (1, -1)-matrices Israel J. Math. 18 1974 353 361
97. E.T.H. Wang On a conjecture of Marcus and Minc Linear and Multilinear Algebra 5 1977 145 148
98. E.T.H. Wang When is the permanent function convex on the set of doubly stochastic matrices? Amer. Math. Monthly 86 1979 119 121
99. I.M. Wanless The Holens-Djokovic conjecture on permanents fails Linear Algebra Appl. 286 1999 273 285
100. I.M. Wanless Maximising the permanent and complementary permanent of (0, 1)-matrices with constant line sum Discrete Math. 205 1999 191 205
101. I.M. Wanless A lower bound on the maximum permanent in Λ n k Linear Algebra Appl. 373 2003 153 167
102. I.M. Wanless, Permanents of matrices of signed ones, Linear and Multilinear Algebra 53 (2005), in press
103. I.M. Wanless, Addendum to Schrijver's work on minimum permanents, Combinatorica, to apper
104. I.M. Wanless, On Minc's Sixth Conjecture, Submitted
105. N. Zagaglia-Salvi Permanents and determinants of circulant (0, 1)-matrices Matematiche (Catania) 39 1984 213 219
106. F.Z. Zhang Notes on Hadamard products of matrices Linear and Multilinear Algebra 25 1989 237 242