The number of non-singleton blocks in Lambda-coalescents with dust

In this article we state and prove a dichotomy, which is satisfied by the number of non-singleton blocks in Lambda-coalescents which have a dust component.
The dichotomy is of a similar type to two well known results, also dichotomies in the behaviour of the Lambda-coalescent, of Pitman (1999) and Schweinsberg (2000). The three dichotomies combine together to give a complete picture of the qualitative behaviour of the singleton/non-singleton blocks of the Lambda-coalescent.