  For all $n > k \ge 1$, we give formulas for the nullity $N(n,k)$ of the $n
\times n$ skew-symmetric Toeplitz band matrix whose first $k$ superdiagonals
have all entries $1$ and whose remaining superdiagonals have all entries $0$.
This is accomplished by counting the number of cycles in certain directed
graphs. As an application, for each fixed integer $z\ge 0$ and large fixed $k$,
we give an asymptotic formula for the percentage of $n > k$ satisfying
$N(n,k)=z$. For the purpose of rapid computation, an algorithm is devised that
quickly computes $N(n,k)$ even for extremely large values of $n$ and $k$.
