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Interpretation of the permanent as the sum of the weights of the injections of an m-element set into an n-element set (m$$\leq n)$$. (English. Russian original) Zbl 0617.05019
Cybernetics 21, 586-591 (1985); translation from Kibernetika 1985, No. 5, 21-24 (1985).
The properties of the permanent are investigated by means of classification of partial mappings of an m-element set into n-element set (m$$\leq n)$$. Ryzer’s and Egorychev’s well-known formulas of the permanent expansion are obtained and their equivalence is shown. The combinatorial content of the polarization formula of polyadditive symmetric function is discovered, and it is proved that the polarization representation is the necessary and sufficient condition of the polyadditive symmetric function.
The author notes that A. M. Kamenetskij has independently obtained these conclusions.
Reviewer: S.S.Agayan
MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 15A15 Determinants, permanents, traces, other special matrix functions
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References:
 [1] G.-C. Rota, ?Baxter algebras and combinatorial identities. II,? Bull. Am. Math. Soc.,75, No. 2, 330?334 (1969). · Zbl 0319.05008 · doi:10.1090/S0002-9904-1969-12158-0 [2] E. A. Bender and J. R. Goldman, ?On the applications of Möbius inversion in combinatorial analysis,? Am. Math. Monthly,82, No. 8, 789?803 (1975). · Zbl 0316.05001 · doi:10.2307/2319793 [3] H. J. Ryser, Combinatorial Mathematics, The Math. Assoc. of America (1963). [4] G. P. Egorychev, ?A family of identities for the permanent function,? Dokl. Akad. Nauk ArmSSR,69, No. 1, 3?7 (1979). · Zbl 0497.15005 [5] G. P. Egorychev, ?A polynomial identity for the permanent,? Mat. Zametki,26, No. 6, 961?964 (1979). [6] G. P. Egorychev, ?New formulas for permanents,? Dokl. Akad. Nauk SSSR,254, No. 4, 784?788 (1980). [7] H. H. Crapo, ?Permanents by Möbius inversion,? J. Combin. Theory,4, No. 2, 198?200 (1968). · Zbl 0162.03201 · doi:10.1016/S0021-9800(68)80043-2 [8] G.-C. Rota, ?On the foundations of combinatorial theory. I. Theory of Möbius functions,? Z. Wahrscheinlichkeitstheorie Verw. Gebiete,2, No. 4, 340?368 (1964). · Zbl 0121.02406 · doi:10.1007/BF00531932 [9] H. Cartan, Differential Calculus, Houghton Mifflin, Boston (1971); Differential Forms, Houghton Mifflin, Boston (1970). · Zbl 0213.37001 [10] A. V. Pogorelov, Geometry [in Russian], Nauka, Moscow (1983). [11] E. Nelson, ?Probability theory and the Euclidean field theory,? in: Constructive Quantum Field Theory, G. Velo and A. Wightman (eds.), Lecture Notes in Physics, No. 25, Springer, Berlin (1973), pp. 94?124. [12] D. Kirby and H. A. Mehran, ?Hilbert functions and the Koszul complex,? J. London Math. Soc.,24, No. 3, 459?466 (1981). · Zbl 0442.13010 · doi:10.1112/jlms/s2-24.3.459 [13] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities [in Russian], Nauka, Leningrad (1980). [14] R. Borges, ?On the principle of inclusion and exclusion,? Period. Math. Hungar.,3, No. 1/2, 149?156 (1973). · Zbl 0249.05007 · doi:10.1007/BF02018470
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