  One of the most useful tools for calculating the decomposition numbers of the
symmetric group is Schaper's sum formula. The utility of this formula for a
given Specht module can be improved by knowing the Schaper Number of the
corresponding partition. Fayers gives a characterisation of those partitions
whose Schaper number is at least two. In this paper we shall demonstrate how
this knowledge can be used to calculate some decomposition numbers before
extending this result with the hope of allowing more decomposition numbers to
be calculated in the future. For $p=2$ we shall give a complete
characterisation of partitions whose Schaper number is at least three, and
those whose Schaper number at least four. We also present a list of necessary
conditions for a partition to have Schaper number at least three for odd primes
and a conjecture on the sufficiency of these conditions.
