Nullities for a class of skew-symmetric Toeplitz band matrices

For all n>k≥1, we give formulas for the nullity N(n,k) of the n×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≥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.