  We describe the minimal free resolution of the ideal of $2 \times 2$
subpermanents of a $2 \times n$ generic matrix $M$. In contrast to the case of
$2 \times 2$ determinants, the $2 \times 2$ permanents define an ideal which is
neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on
the Gr\"obner basis of an ideal of $2 \times 2$ permanents of a generic matrix
with our previous work connecting the initial ideal of $2 \times 2$ permanents
to a simplicial complex. The main technical tool is a spectral sequence arising
from the Bernstein-Gelfand-Gelfand correspondence.
