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Iskovskikh Seminar
September 10, 2020 18:00–19:30, Moscow, Steklov Mathematical Institute, room 540

On the values of permanent function

A. È. Guterman

$$\det A= \sum_{\sigma\in { S}_n} (-1)^{\sigma} a_{1\sigma(1)}\cdots a_{n\sigma(n)} \quad {\mathrm and } \quad {\mathrm per} A= \sum_{\sigma\in { S}_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)},$$
here $A=(a_{ij})\in M_n({\mathbb F})$ is $n\times n$ matrix over a certain field ${\mathbb F}$ and $S_n$ denotes the symmetric group on $\{1,\ldots, n\}$. However the properties of permanent function are much more involved. For example, by Gauss elimination algorithm determinant can be computed in polynomial time, but the existence of polynomial algorithm to compute permanent is an open problem. Due to this reason series of open problems concerning the values of permanent and relations between permanent and determinant are actual.