Abstract:
The permanent of a multidimensional matrix is the sum of the products of entries over all diagonals. In this survey, we consider the basic properties of the multidimensional permanent, sufficient conditions for its positivity, available upper bounds, and the specifics of the permanents of polystochastic matrices. We prove that the number of various combinatorial objects can be expressed via multidimensional permanents. Special attention is paid to the number of $1$-factors of uniform hypergraphs and the number of transversals in Latin hypercubes. Tabl. 1, bibliogr. 63.
Keywords:permanent, multidimensional matrix, stochastic matrix, polystochastic matrix, transversal in a Latin hypercube, $1$-factor of a uniform hypergraph.
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