Let \(e_k(x_1,\ldots ,x_\ell )\) be an elementary symmetric polynomial and let \(\lambda =(\lambda _1,\ldots ,\lambda _\ell )\) be an integer partition. Define \({{\,\textrm{pre}\,}}_k(\lambda )\) to be the partition whose parts are the summands in the evaluation \(e_k(\lambda _1,\ldots ,\lambda _\ell )\) . The study of such partitions was initiated by Ballantine, Beck, and Merca who showed (among other things) that \({{\,\textrm{pre}\,}}_2\) is injective as a map on binary partitions of n. In the present work, we derive a host of identities involving the sequences which count the number of parts of a given value in the image of \({{\,\textrm{pre}\,}}_2\) . These include generating functions, explicit expressions, and formulas for forward differences. We generalize some of these to d-ary partitions and explore connections with color partitions. Our techniques include the use of generating functions and bijections on rooted partitions. We end with a list of conjectures and a direction for future research.
