Factor-set of binary matrices and Fibonacci numbers

The article discusses the set of square n×nn×n binary matrices with the same number of 1’s in each row and each column. An equivalence relation on this set is introduced. Each binary matrix is represented using ordered n-tuples of natural numbers. We are looking for a formula which calculates the number of elements of each factor-set by the introduced equivalence relation. We show a relationship between some particular values of the parameters and the Fibonacci sequence.