On an Involution Concerning Pairs of Polynomials over F2

From the work of S. Corteel et al. (1998, J. Combin. Theory Ser. A82, 186–192), the number of coprime m-tuples of monic polynomials of degree n over Fq is equal to qnm−q(n−1) m+1. In particular, among the ordered pairs of polynomials of degree n over F2 there are as many relatively prime as non-relatively prime ones. We give an involution that proves this result.