  Many of the known complemented subspaces of L_p have realizations as sequence
spaces. In this paper a systematic approach to defining these spaces which uses
partitions and weights is introduced. This approach gives a unified description
of many well-known complemented subspaces of L_p. It is proved that the class
of spaces with such norms is stable under (p,2) sums. By introducing the notion
of an envelope norm, we obtain a necessary condition for a Banach sequence
space with norm given by partitions and weights to be isomorphic to a subspace
of L_p. Using this we define a space Y_n with norm given by partitions and
weights with distance to any subspace of L_p growing with n. This allows us to
construct an example of a Banach space with norm given by partitions and
weights which is not isomorphic to a subspace of L_p.
