  Trait allocations are a class of combinatorial structures in which data may
belong to multiple groups and may have different levels of belonging in each
group. Often the data are also exchangeable, i.e., their joint distribution is
invariant to reordering. In clustering---a special case of trait
allocation---exchangeability implies the existence of both a de Finetti
representation and an exchangeable partition probability function (EPPF),
distributional representations useful for computational and theoretical
purposes. In this work, we develop the analogous de Finetti representation and
exchangeable trait probability function (ETPF) for trait allocations, along
with a characterization of all trait allocations with an ETPF. Unlike previous
feature allocation characterizations, our proofs fully capture
single-occurrence "dust" groups. We further introduce a novel constrained
version of the ETPF that we use to establish an intuitive connection between
the probability functions for clustering, feature allocations, and trait
allocations. As an application of our general theory, we characterize the
distribution of all edge-exchangeable graphs, a class of recently-developed
models that captures realistic sparse graph sequences.
