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On the values of permanent function A. È. Guterman |
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Abstract: Two matrix functions determinant and permanent are important in algebra, combinatorics, and their applications and look quite similar: $$ \det A= \sum_{\sigma\in { S}_n} (-1)^{\sigma} a_{1\sigma(1)}\cdots a_{n\sigma(n)} \quad {\mathrm and } \quad {\mathrm per}\, A= \sum_{\sigma\in { S}_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}, $$ here In the talk we discuss the results of a series of joint works with M.V. Budrevich, G. Dolinar, B. Kuzma, I.A. Spiridonov, K.A. Taranin, devoted to our recent progress in the following directions: Polya problem of permanent-determinant conversion; Brualdi-Newman problem of non-realizable values of permanent of (0,1) matrices; the positive solution of Wang-Kraüter problem on exact upper bound for permanent of (-1,1) matrices. |