Values of the permanent function on multidimensional $(0,1)$-matrices

We study the range of the permanent function for the multidimensional matrices of $0$ and $1$. The main result is a multidimensional version for the Brualdi–Newman upper bound on the consecutive values of the permanent (1965). Moreover, we deduce a formula for the permanent of the multidimensional $(0,1)$-matrices through the number of partial zero diagonals. Using the formula, we evaluate the permanents of the $(0,1)$-matrices with a few zeros and estimate the permanents of the matrices whose all zero entries are located in several orthogonal hyperplanes. We consider some divisibility properties of the permanent and illustrate the results by studying the values of the permanent for the $3$-dimensional $(0,1)$-matrices of order $3$.