  We prove that the v-decomposition number $d_{\lambda\mu}(v)$ is an even or
odd polynomial according to whether the partitions $\lambda$ and $\mu$ have the
same relative sign (or parity) or not. We then use this result to verify
Martin's conjecture for weight 3 blocks of symmetric group algebras -- that
these blocks have the property that their projective (indecomposable) modules
have a common radical length 7.
