Bulletin of the American Mathematical Society

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Book Information:

Author: Henryk Minc
Title: Permanents
Additional book information: Encyclopedia of Mathematics and its Applications (Gian-Carlo Rota, Editor), Volume 6, Addison-Wesley, Reading, Mass., 1978, xviii + 205 pp., $21.50.

1.

T. Bang, Matrixfunktioner som med et numerisk lille deficit viser v.d. Waerdens permanenthypothese, Proc. Scandinavian Congress, Turkku, 1976.

2.

J. P. M. Binet, Mémoire sur un système de formules analytiques, et leur application à des considérations géométriques, J. Ec. Polyt. 9 (1812); Cah. 16, 280-302.

Garrett Birkhoff, Three observations on linear algebra, Univ. Nac. Tucumán. Revista A. 5 (1946), 147–151 (Spanish). MR 0020547

L. M. Brègman, Certain properties of nonnegative matrices and their permanents, Dokl. Akad. Nauk SSSR 211 (1973), 27–30 (Russian). MR 0327788

J. L. Brenner, Relations among the minors of a matrix with dominant principal diagonal, Duke Math. J. 26 (1959), 563–567. MR 110722

J. L. Brenner and R. A. Brualdi, Eigenschaften der Permanentefunktion, Arch. Math. (Basel) 18 (1967), 585–586 (German). MR 220743, DOI 10.1007/BF01898864

Richard A. Brualdi and Peter M. Gibson, Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function, J. Combinatorial Theory Ser. A 22 (1977), no. 2, 194–230. MR 437562, DOI 10.1016/0097-3165(77)90051-6

E. R. Caianiello, Combinatorics and renormalization in quantum field theory, Frontiers in Physics, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0464973

9.

A. L. Cauchy, Mémoire sur les fonctions . . . , J. Ec. Polyt. 10 (1812); Cah. 17, 29-112, Oeuvres (2) i.

P. J. Eberlein, Remarks on the van der Waerden conjecture. II, Linear Algebra Appl. 2 (1969), 311–320. MR 246891, DOI 10.1016/0024-3795(69)90033-0

P. J. Eberlein and Govind S. Mudholkar, Some remarks on the van der Waerden conjecture, J. Combinatorial Theory 5 (1968), 386–396. MR 231835

Thomas H. Foregger, An upper bound for the permanent of a fully indecomposable matrix, Proc. Amer. Math. Soc. 49 (1975), 319–324. MR 369385, DOI 10.1090/S0002-9939-1975-0369385-4

Shmuel Friedland, Matrices satisfying the van der Waerden conjecture, Linear Algebra Appl. 8 (1974), no. 6, 521–528. MR 453770, DOI 10.1016/0024-3795(74)90086-x

Shmuel Friedland, A lower bound for the permanent of a doubly stochastic matrix, Ann. of Math. (2) 110 (1979), no. 1, 167–176. MR 541335, DOI 10.2307/1971250

D. J. Hartfiel, A simplified form for nearly reducible and nearly decomposable matrices, Proc. Amer. Math. Soc. 24 (1970), 388–393. MR 252415, DOI 10.1090/S0002-9939-1970-0252415-3

D. J. Hartfiel, A lower bound for the permanent on a special class of matrices, Canad. Math. Bull. 17 (1974), no. 4, 529–530. MR 384830, DOI 10.4153/CMB-1974-093-2

J. M. Hammersley, An improved lower bound for the multidimensional dimer problem, Proc. Cambridge Philos. Soc. 64 (1968), 455–463. MR 237350, DOI 10.1017/s030500410004305x

18.

J. M. Hammersley et al., Negative finding for the three-dimensional dimer problem, J. Mathematical Phys. 10 (1969), 433-446.

19.

G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, London, 1934.

W. B. Jurkat and H. J. Ryser, Matrix factorizations of determinants and permanents, J. Algebra 3 (1966), 1–27. MR 195879, DOI 10.1016/0021-8693(66)90016-0

W. B. Jurkat and H. J. Ryser, Term ranks and permanents of nonnegative matrices, J. Algebra 5 (1967), 342–357. MR 215740, DOI 10.1016/0021-8693(67)90044-0

C. H. C. Little, A characterization of convertible (0,1)-matrices, J. Combinatorial Theory Ser. B 18 (1975), 187–208. MR 424583, DOI 10.1016/0095-8956(75)90048-9

Marvin Marcus and Henryk Minc, On the relation between the determinant and the permanent, Illinois J. Math. 5 (1961), 376–381. MR 147488

Marvin Marcus and Henryk Minc, Permanents, Amer. Math. Monthly 72 (1965), 577–591. MR 177000, DOI 10.2307/2313846

Marvin Marcus and Henryk Minc, Generalized matrix functions, Trans. Amer. Math. Soc. 116 (1965), 316–329. MR 194445, DOI 10.1090/S0002-9947-1965-0194445-9

Marvin Marcus and Morris Newman, On the minimum of the permanent of a doubly stochastic matrix, Duke Math. J. 26 (1959), 61–72. MR 104679

Marvin Marcus and Morris Newman, Inequalities for the permanent function, Ann. of Math. (2) 75 (1962), 47–62. MR 136628, DOI 10.2307/1970418

Henryk Minc, Upper bounds for permanents of $(0,\,1)$-matrices, Bull. Amer. Math. Soc. 69 (1963), 789–791. MR 155843, DOI 10.1090/S0002-9904-1963-11031-9

Henryk Minc, On lower bounds for permanents of $(0,\,1)$ matrices, Proc. Amer. Math. Soc. 22 (1969), 117–123. MR 245585, DOI 10.1090/S0002-9939-1969-0245585-6

30.

T. Muir, On a class of permanent symmetric functions, Proc. Roy. Soc. Edinburgh 11 (1882), 409-418.

31.

R. F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc. Edinburgh Math. Soc. 21 (1903), 144-157.

J. K. Percus, Combinatorial methods, Courant Institute of Mathematical Sciences, New York University, New York 1969. Notes recorded by Ora Engelberg Percus. MR 0263644

33.

G. Pólya, Aufgabe 424, Arch. Math. Phys. 20 (1913), 271.

Herbert John Ryser, Combinatorial mathematics, The Carus Mathematical Monographs, No. 14, Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York, 1963. MR 0150048

D. W. Sasser and M. L. Slater, On the inequality $\sum x_{i}y_{i}\geqq (1/n)\sum x_{i}\cdot \sum y_{i}$ and the van der Waerden permanent conjecture, J. Combinatorial Theory 3 (1967), 25–33. MR 217099

A. Schrijver, A short proof of Minc’s conjecture, J. Combinatorial Theory Ser. A 25 (1978), no. 1, 80–83. MR 491216, DOI 10.1016/0097-3165(78)90036-5

I. Schur, Über endliche Gruppen und Hermitesche Formen, Math. Z. 1 (1918), no. 2-3, 184–207 (German). MR 1544291, DOI 10.1007/BF01203611

Richard Sinkhorn, Concerning a conjecture of Marshall Hall, Proc. Amer. Math. Soc. 21 (1969), 197–201. MR 241440, DOI 10.1090/S0002-9939-1969-0241440-6

Richard Sinkhorn and Paul Knopp, Problems involving diagonal products in nonnegative matrices, Trans. Amer. Math. Soc. 136 (1969), 67–75. MR 233830, DOI 10.1090/S0002-9947-1969-0233830-7

40.

B. L. van der Waerden, Aufgabe 45, Jber. Deutsch. Math.-Verein 35 (1926), 117.

M. Voorhoeve, A lower bound for the permanents of certain $(0,\,1)$-matrices, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 1, 83–86. MR 528221

1.

T. Bang, Matrixfunktioner som med et numerisk lille deficit viser v.d. Waerdens permanenthypothese, Proc. Scandinavian Congress, Turkku, 1976.

2.

J. P. M. Binet, Mémoire sur un système de formules analytiques, et leur application à des considérations géométriques, J. Ec. Polyt. 9 (1812); Cah. 16, 280-302.

3.

G. Birkhoff, Très observaciones sobre el algebra lineal, Univ. Nac. Tucumán Rev. Ser. A 5 (1946), 147-151. MR 0020547

4.

L. M. Brégman, Certain properties of nonnegative matrices and their permanents, Soviet Math. Dokl. 14 (1973), 945-949. MR 327788

5.

J. L. Brenner, Relations among the minors of a matrix with dominant principal diagonal, Duke Math. J. 26 (1959), 563-568. MR 110722

6.

J. L. Brenner and R. A. Brualdi, Eigenshaften der Permanente funktionen, Arch. Math. 18 (1967), 585-586. MR 220743

7.

R. A. Brualdi and P. M. Gibson, The convex polyhedron of doubly stochastic matrices: I. Applications of the permanent function, J. Combinatorial Theory Ser. A 22 (1977), 194-230. MR 437562

8.

E. R. Caianiello, Combinatorics and renormalisation in quantum field theory, New York, 1974. MR 464973

9.

A. L. Cauchy, Mémoire sur les fonctions . . . , J. Ec. Polyt. 10 (1812); Cah. 17, 29-112, Oeuvres (2) i.

10.

P. J. Eberlein, Remarks on the van der Waerden conjecture. II, Linear Algebra and Appl. 2 (1969), 311-320. MR 246891

11.

P. J. Eberlein and G. S. Mudholkar, Some remarks on the van der Waerden conjecture, J. Combinatorial Theory 5 (1968), 386-396. MR 231835

12.

T. H. Foregger, An upper bound for the permanent of a fully indecomposable matrix, Proc. Amer. Math. Soc. 49 (1975), 319-324. MR 369385

13.

S. Friedland, Matrices satisfying the van der Waerden conjecture, Linear Algebra and Appl. 8 (1974), 521-528. MR 453770

14.

S. Friedland, A lower bound for the permanent of a doubly stochastic matrix, Linear and Multilinear Algebra (to appear) MR 541335

15.

D. J. Hartfiel, A simplified form for nearly reducible and nearly decomposable matrices, Proc. Amer. Math. Soc. 24 (1970), 388-393. MR 252415

16.

D. J. Hartfiel, A lower bound for the permanent on a special class of matrices, Canad. Math. Bull. 17 (1974), 529-530. MR 384830

17.

J. M. Hammersley, An improved lower bound for the multidimensional dimer problem, Proc. Cambridge Philos. Soc. 64 (1968), 455-463. MR 237350

18.

J. M. Hammersley et al., Negative finding for the three-dimensional dimer problem, J. Mathematical Phys. 10 (1969), 433-446.

19.

G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, London, 1934.

20.

W. B. Jurkat and H. J. Ryser, Matrix factorizations of determinants and permanents, J. Algebra 3 (1966), 1-27. MR 195879

21.

W. B. Jurkat and H. J. Ryser, Term ranks and permanents of nonnegative matrices, J. Algebra 5 (1967), 342-357. MR 215740

22.

C. H. C. Little, A characterization of convertible (0, 1)-matrices, J. Combinatorial Theory 18 (1975), 187-208. MR 424583

23.

M. Marcus and H. Minc, On the relation between the determinant and the permanent, Illinois J. Math. 5 (1961), 376-381. MR 147488

24.

M. Marcus and H. Minc, Permanents, Amer. Math. Monthly 72 (1965), 577-591. MR 177000

25.

M. Marcus and H. Minc, Generalized matrix functions, Trans. Amer. Math. Soc. 116 (1965), 316-329. MR 194445

26.

M. Marcus and M. Newman, On the minimum of the permanent of a doubly stochastic matrix, Duke Math. J. 26 (1959), 61-72. MR 104679

27.

M. Marcus and M. Newman, Inequalities for the permanent function, Ann. of Math. (2) 675 (1962), 47-62. MR 136628

28.

H. Minc, Upper bounds for permanents of (0, 1)-matrices, Bull. Amer. Math. Soc. 69 (1963), 789-791. MR 155843

29.

H. Minc, On lower bounds for permanents of (0, 1)-matrices, Proc. Amer. Math. Soc. 22 (1969), 117-123. MR 245585

30.

T. Muir, On a class of permanent symmetric functions, Proc. Roy. Soc. Edinburgh 11 (1882), 409-418.

31.

R. F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc. Edinburgh Math. Soc. 21 (1903), 144-157.

32.

J. K. Percus, Combinatorial methods, Courant Institute of Mathematical Sciences, New York, 1969. MR 263644

33.

G. Pólya, Aufgabe 424, Arch. Math. Phys. 20 (1913), 271.

34.

H. J. Ryser, Combinatorial mathematics, Carus Math. Monograph No. 14, Math. Assoc. Amer., 1963. MR 150048

35.

D. W. Sasser and M. L. Slater, On the inequality $\sum x\sbiy\sbi\geqq (1/n)\sum x\sbi·\sum y\sbi$ and the van der Waerden conjecture, J. Combinatorial Theory 3 (1967), 25-33. MR 217099

36.

A. Schrijver, A short proof of Minc's conjecture, J. Combinatorial theory Ser A. 25 (1978), 80-83. MR 491216

37.

I. Schur, Über endliche Gruppen und Hermitesche Formen, Math. Z. 1 (1918), 184-207. MR 1544291

38.

R. Sinkhorn, Concerning a conjecture of Marshall Hall, Proc. Amer. Math. Soc. 21 (1969), 197-201. MR 241440

39.

R. Sinkhorn and P. Knopp, Problems involving diagonal products in nonnegative matrices, Trans. Amer. Math. Soc. 136 (1969), 67-75. MR 233830

40.

B. L. van der Waerden, Aufgabe 45, Jber. Deutsch. Math.-Verein 35 (1926), 117.

41.

M. Voorhoeve (to appear). MR 528221