The multiplicity of an irreducible general linear group representation in a composition of irreducible general linear group representations are known as plethysm coefficients. One very difficult and long standing open problem in combinatorial representation theory is to give a positive or combinatorial formula for plethysm coefficients. In this talk I will present some new results about the algebra of uniform block permutations (sometimes known as "the party algebra") and explain how they are related to the operation of plethysm and might be used to help resolve this problem. This is joint work with Rosa Orellana, Anne Schilling and Franco Saliola.