  Given a partition $\{I_1,\ldots,I_k\}$ of $\{1,\ldots,n\}$, let
$(X_1,\ldots,X_n)$ be random vector with each $X_i$ taking values in an
arbitrary measurable space $(S,\mathscr{S})$ such that their joint law is
invariant under finite permutations of the indexes within each class $I_j$.
Then, it is shown that this law has to be a signed mixture of independent laws
and identically distributed within each class $I_j$. The representation is
unique if and only if the set of these signed measures is weakly compact.
  We provide a necessary condition for the existence of a nonnegative directing
measure. This is related to the notions of infinite extendibility and
reinforcement. In the special case where $(X_1,\ldots,X_n)$ is an exchangeable
sequence of $\{0,1\}$-valued random variables, the directing measure can be
chosen nonnegative if and only if two effectively computable matrices are
positive semi-definite.
