We present a Diagnostic dataset of IA References in a Pentomino domain (Pento-DIARef) that ties extensional and intensional definitions more closely together, insofar as the latter is the generative process creating the former. We create a novel synthetic dataset of examples that pairs visual scenes with generated referring expressions; examine two variants of the dataset, representing two different ways to exemplify the underlying task; and evaluate an LSTM-based baseline, a transformer and a modified version with region embeddings on them. See https://github.com/clp-research/pento-diaref for more information.