Representing Integers in the Binary Number System as Permanents of Certain Matrices

Abstract

The permanent of an $m$-by-$n$ matrix $A$ is the sum of all possible products of $m$ elements from $A$ with the property that the elements in each of the products lie on different lines of $A$. This scalar valued function of the matrix $A$ occurs throughout the combinatorial literature in connection with various enumeration and extremal problems. In this note, we construct a $(0,1)$-matrix with a prescribed permanent, $1$, $2$, $\ldots$, $2^{n-1}$.