Permanents of d-dimensional matrices
For an \(n_ 1\times n_ 2\times...\times n_ d\) matrix \(A=(A_{i_ 1i_ 2...i_ d})\) the permanent function is defined by \(per A=\sum \prod^{n_ 1}_{i=1}A_{i\sigma_ 2(i)\sigma_ 3(i)...\sigma_ d(i)},\) where the summation ranges over all one-to-one functions \(\sigma_ k\) from \(\{1,2,...,n_ 1\}\) to \(\{1,2,...,n_ k\}\), \(k=2,3,...,d\). Necessarily for each k, \(n_ k\geq n_ 1\). A particular interest is given to those matrices which are nonnegative and have constant hyperplane sums. Many of the properties of permanents of two- dimensional matrices extend to the higher dimensional matrices.
scientific article; zbMATH DE number 3993763
scientific article