We study the combinatorial structure of the state-space of non-deterministic fork-join processes. As a first step we establish a link between concurrent programs and a class of combinatorial structures based on the notion of increasing labelling. Beyond the theory, our goal is to develop algorithms and tools for the statistical analysis of the process behaviours. In the second part we develop and experiment two efficient random sampling algorithms serving as basic building blocks for state-space exploration. One is a uniform random sampler of bounded executions, providing a good default exploration strategy, and the other is a uniform sampler of execution prefixes allowing to bias the exploration in a controlled manner. The fundamental characteristic of these algorithms is that they work on the control graph of the programs and do not require the construction of its state-space, thus providing a way to tackle the infamous state explosion problem.