Elementary proof for a Van der Waerdenʹs conjecture and related theorems

A well-known conjecture of Van der Waerden says that for the permanent Per A of an n × n doubly stochastic matrix A we have (3.0), with equality if and only if all entries of the matrix A equal to n−1. In 1977 [1], the author proved that if A is an n × n doubly stochastic matrix, and p ≥ 0, q ≥ 0, p + Q = 1, then (2.0) holds with equality if and only if all entries of the matrix A equal to n−1. In this paper, we show that (3.0) and (2.0) are equivalent. On the basis of this equivalence one can say that the equivalent of the Van der Waerdenʹs conjecture was solved already in 1977. A further subject of the paper is to show similar equivalence theorems concerning permanents of doubly stochastic matrices, moreover a refinement of the Van der Waerdenʹs theorem. A separate section deals with the probabilistic interpretation of some previous results.