On the number of different permanents of some sparse (0,1) circulant matrices

Starting from known results about the number of possible values for the permanents of $(0,1)$-circulant matrices with three nonzero entries per row, and whose dimension $n$ is prime, we prove corresponding results for $n$ power of a prime, $n$ product of two distinct primes, and $n=2\cdot 3^h$. Supported by some experimental results, we also conjecture that the number of different permanents of $n\times n$ $(0,1)$-circulant matrices with $k$ nonzero per row is asymptotically equal to $n^{k-2}/k!+O(n^{k-3}).$