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The Permanent Function

Published online by Cambridge University Press:  20 November 2018

Marvin Marcus and
F. C. May
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Let X be an n-square matrix with elements in a field F. The permanent of X is defined by

1.1

where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962, pp. 177 - 189
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Aitken, A. C., Determinants and matrices, (5th ed.; Edinburgh: Oliver and Boyd, 1948).Google Scholar
2. Dieudonné, J., Sur une généralisation du groupe orthogonal à quatre variables, Archiv d. Math., 1 (1948), 282–287.Google Scholar
3. Frobenius, G., Ueber die Darstellung der endlichen Gruppen durch linearc Substitutionen, Sitzungber. der Berliner Akademie, 994–1015 (§7).Google Scholar
4. Jacob, H. G., Coherence invariant mappings on Kronecker products, Amer. J. Math., 77 (1955), 177–189.Google Scholar
5. Kantor, S., Théorie der Äquivalenz von linearen ∞-Schartn bilinearer Formen, Sitzungber. der Miinchener Akademie (1897), 367–381 (§2).Google Scholar
6. Kônig, D., Théorie der Graphen (New York: Chelsea, 1950), 238.Google Scholar
7. Marcus, Marvin and May, F. C., On a theorem of I. Schur concerning matrix transformations; Archiv der Mathematik, 11 (1960), 401–404.Google Scholar
8. Marcus, Marvin and Moyls, B. N., Linear transformations on algebras of matrices, Can. J. Math., 11 (1959), 61–66.Google Scholar
9. Marcus, Marvin and Newman, M., Permanents of doubly stochastic matrices, Proc. Symposia Applied Math., 10, Amer. Math. Soc. (1960), 169–174.Google Scholar
10. Marcus, Marvin and Purves, R., Linear transformations on algebras of matrices: the invariance of the elementary symmetric functions, Can. J. Math., 11 (1959), 383–396.Google Scholar
11. Morita, K., Schwarz's lemma in a homogeneous space of higher dimensions, Jap. J. Math., 19 (1944), 45–56.Google Scholar
12. Pólya, G., Aufgabe 424, Archiv d. Math. u. Phys., 20 (3), 271.Google Scholar
13. Ryser, H. J., Compound and induced matrices in combinatorial analysis, Proc. Symposia Applied Math., 10, Amer. Math. Soc. (1960), 166.Google Scholar
14. Schur, I., Einige Bemerkungen zur Determinantentheorie, Sitzungber. der Preussischen Akademie der Wissenschaften zu Berlin, 25 (1925), 454–463.Google Scholar