A note on the signature representations of the symmetric groups

For a partition {\lambda} and a prime p, we prove a necessary and sufficient condition for there exists a composition {\delta} such that {\delta} can be obtained from {\lambda} after rearrangement and all the partial sums of {\delta} are not divisible by p.Comment: This is a substantially revised version of previous ones. In the third section, we calculate some explicit p-Kostka number