In this paper, I discuss undergraduate students' engagement in basic Python programming while solving combinatorial problems. Students solved tasks that were designed to involve programming, and they were encouraged to engage in activities of prediction and reflection. I provide data from two paired teaching experiments, and I outline how the task design and instructional interventions particularly supported students' combinatorial reasoning. I argue that emergent computational representations and the prompts for prediction and reflection were especially useful in supporting students' reasoning about fundamental combinatorial ideas. I argue that this particular mathematical example informs broader notions of disciplinary reflexivity and representational heterogeneity, providing insight into computational thinking practices in the domain of mathematics. Ultimately, I aim to explore the nature of computing and enumeration, shedding light on why the two disciplines are particularly well-suited to support each other. I conclude with implications and avenues for future research.