On the parity of permanents of circulant matrices Original Research Article

Starting from previous results concerning determinants and permanents of (0,1) circulant matrices, and using theory on finite fields with characteristic 2, we first see (denoting by A the n×n (0,1) circulant matrix whose 1’s in the first row are exactly in positions i1,i2,…,ik) that Per(A) is even iff gcd(xi1+xi2+cdots, three dots, centered+xik,1+xn)≠1 in Z2[x]. We then derive some consequences for specific classes of primes n: in particular, when n is a prime such that the group image is generated by the residue class 2, we see that for k≠n and k odd the value Per(A) is always odd. Similarly, for n>7 and n prime with the group of squares in image generated by the class 2, we see that for k=3 the value Per(A) is always odd. Contrary to such cases, we find that when n is a Mersenne prime greater than 3, there are a significant number of circulants A of the above form with k odd and Per(A) even.