The clump giants in old and middle-aged open clusters provide a sample of stars whose evolutionary state can be clearly identified as core helium burning (Cannon 1970). The evolutionary state of the Ba II stars, on the other hand, is at present undetermined. The enrichment of s-process elements in the atmospheres of these peculiar stars suggests that the stars have at least passed through double shell burning, where the heavy elements may be produced and mixed to the surface. The Ba II stars are, however, too faint to be associated with this phase of stellar evolution; the absolute magnitudes and temperatures are consistent with the core helium burning phase. The barium stars should then occur in the clumps of Population I clusters in similar proportion to their numbers among the field giants.