The Range of the Determinant Function on the Set of $nxn$ $(0,1)$-Matrices

Abstract

As stated in $[2]$, there is a conjecture that the determinant function maps the set of $n\times n$ $(0,1)$-matrices onto a set of consecutive integers for any given $n$. While this is true for $n\le 6$, it is shown to be false for $n=7$. In particular, there is no $7\times 7$ determinant in the range $28–31$, but there is one equal to $32$. Then the more general question of the range of the determinant function for all $n$ is discussed. A lower bound is given on the largest string of consecutive integers centered at $0$, each of which is a determinant of an $n\times n$ $(0,1)$-matrix.