  For any prime p, we construct, and simultaneously count, all of the complex
Specht modules in a given p-block of the symmetric group which remain
irreducible when reduced modulo p. We call the Specht modules with this
property p-irreducible modules. Recently Fayers has proven a conjecture of
James and Mathas that provides a characterization of the partitions that
correspond to the p-irreducible modules. In this paper we present a method for
decomposing the partitions corresponding to p-irreducible modules, and we use
this decomposition to construct and count all of the partitions corresponding
to p-irreducible Specht modules in a given block.
