What is the maximum possible value of determinant of a matrix whose entries either 0 or 1?

My question is simply the title:

What is the maximum possible value of determinant of a matrix whose entries either 0 or 1 ?

Solution 1:

Quoting my question in another thread:

In fact, I don't even know how large the determinant of a 0-1 matrix can be. The Hadamard's bound for the absolute determinant of an $n\times n$ 0-1 matrix is $\frac{(n+1)^{(n+1)/2}}{2^n}$ (online ref. 1 and ref. 2), and the bound is sharp if and only if there exists a Hadamard matrix of order $n+1$. Yet, to my knowledge, there is no known sharp upper bound for the absolute determinant of a general $n\times n$ 0-1 matrix.

Solution 2:

A few examples of $\{0,1\}$-matrices (with the largest determinants $-$ according to OEIS-A003432):

$n=2$: $\quad\det\left( \begin{array}{cc} \bf{1} & 0 \\ \bf{1} & \bf{1} \\ \end{array} \right) =1;$

$n=3$: $\quad\det\left( \begin{array}{ccc} \bf{1} & 0 & \bf{1} \\ \bf{1} & \bf{1} & 0 \\ 0 & \bf{1} & \bf{1} \\ \end{array} \right) =2;$

$n=4$: $\quad\det\left( \begin{array}{cccc} \bf{1} & 0 & \bf{1} & 0 \\ \bf{1} & \bf{1} & 0 & \bf{1} \\ 0 & \bf{1} & \bf{1} & 0 \\ 0 & 0& \bf{1} & \bf{1} \\ \end{array} \right) =3;$

$n=5$: $\quad\det\left( \begin{array}{ccccc} \bf{1} & 0 & \bf{1} & 0 & 0\\ \bf{1} & \bf{1} & 0 & \bf{1} & 0 \\ 0 & \bf{1} & \bf{1} & 0 &\bf{1}\\ 0 & 0 & \bf{1} & \bf{1} & 0 \\ \bf{1} & 0 & 0 & \bf{1} & \bf{1} \\ \end{array} \right) =5;$

$n=6$: $\quad\det\left( \begin{array}{ccccc} \bf{1} & 0 & \bf{1} & 0 & 0 & 0\\ \bf{1} & \bf{1} & 0 & \bf{1} & 0 & 0 \\ 0 & \bf{1} & \bf{1} & 0 &\bf{1} & 0\\ 0 & 0 & \bf{1} & \bf{1} & 0 & \bf{1}\\ \bf{1} & 0 & 0 & \bf{1} & \bf{1} & 0\\ \bf{1} & \bf{1} & 0 & 0 & \bf{1} & \bf{1} \\ \end{array} \right) =9;$

$n=7$: $\quad\det\left( \begin{array}{ccccc} \bf1 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 \\ \bf1 & \bf1 & 0 & \bf1 & 0 & 0 & \bf1 \\ \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 & 0 \\ 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 \\ 0 & 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 \\ \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 & 0 \\ 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 \\ \end{array} \right) =32;$

$n=8$: $\quad\det\left( \begin{array} \bf1 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & 0 \\ \bf1 & \bf1 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 \\ \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 & 0 & \bf1 \\ 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 & 0 \\ 0 & 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 \\ \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 \\ 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 & 0 \\ 0 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 \\ \end{array} \right) =56;$