First degree cohomology of Specht modules over fields of odd prime characteristic

For a field k of odd prime characteristic, we give an easy, but far-reaching sufficient combinatorial condition for a partition λ   indicating that the first cohomology H1(Sn,Sλ)H1(Sn,Sλ) of the corresponding Specht module is trivial. Based on this condition, an infinite set of partitions λ   is determined, for which the first cohomology groups H1(Sn,Sλ)H1(Sn,Sλ) are one-dimensional.