Spectrum of Permanent’s Values and Its Extremal Magnitudes in and

Let denote the class of square matrices containing in each row and in each column exactly 1’s. The minimal value of , for which the behavior of the permanent in is not quite studied, is . We give a simple algorithm for calculation of upper magnitudes of permanent in and consider some extremal problems in a generalized class , the matrices of which contain in each row and in each column nonzero elements , , and and zeros. 1. Introduction The definition of permanent of a square matrix of order is , where sum is over all permutations of numbers . This definition is very combinatorial. For example, if is -matrix, then is the number of arrangements of nonattacking each other rooks on positions of 1’s of . Therefore, the most natural applications of the permanent are in combinatorics: enumerating the Latin rectangles and squares, permutations with restricted positions, and different problems in the theory of graphs. Many interesting examples of applications of the permanent one can find in [1], chapter 8 (and not only in Mathematics!). To applications of the permanent there promoted celebrate proofs of the best known van der Waerden’s and Minc’s conjectures by Egorychev [2], Falikman [3], and Brègman [4], respectively. For example, Alon and Friedland [5] excellently used the Minc-Bregman inequality for permanent of -matrices for finding the maximum number of perfect matchings in graphs with a given degree sequence. Let denote the class of square matrices containing in each row and in each column exactly 1’s. If , then matrix is doubly stochastic, since all its row and column sums equal 1. Therefore, -matrices are also called doubly stochastic (0,1)-matrices. These matrices have especially simple and attractive structure and have many applications. It is important that especially class in permanent’s history was a nontrivial test area for different researches of the permanent. For example, one can mention a remarkable Merriell research [6] of the maximum of permanent on class . An author’s research [7] of the permanent on -matrices (see Section 9) has led to the disproof of an important Balasubramanian conjecture [8] which would yield a full proof of 1967 Ryser hypothesis [9] on transversals of Latin squares. In the present paper we consider many other problems for . We study also the following natural generalization of . For given real or complex nonzero numbers , , and , denote by the class of square matrices containing every number from exactly one time in each row and in each column, such that the other elements are 0’s. It is clear that, if and , then the