Parity considerations in the number of parts
Thomas Y. He, H. X. Huang, Y. X. Xie, T. T. Zou
TL;DR
The paper addresses parity questions for parts in overpartitions, specifically the parity of the number of non-overlined parts of size $\leq\widetilde{LO}(\pi)$ and of overlined parts of size $\leq\widetilde{SN}(\pi)$, alongside the parity of even parts above the smallest odd part in partitions. It develops four new parity-difference identities for overpartitions, mirroring and extending prior results, with explicit generating-functions and corollaries. The authors supply both analytic and combinatorial proofs, including involution-based frameworks that connect parity counts to classical partition statistics such as $D(n)$, $D_e(n)$, and $p(n)$, and they present a companion set of results under the label 'new-A-B-thm' with multiple equivalent expressions. Overall, the work deepens the understanding of parity phenomena in partition theory and yields nonnegativity results and structured $q$-series identities suitable for further combinatorial interpretation and generalization.
Abstract
Recently, Chen, He, Hu and Xie considered the parity of the number of non-overlined (resp. overlined) parts of size greater than or equal to the size of the smallest overlined (resp. non-overlined) part in an overpartition. In this article, we investigate the parity of the number of non-overlined (resp. overlined) parts of size less than or equal to the size of the largest overlined (resp. non-overlined) part in an overpartition. We also study the parity of the number of even parts greater than the smallest odd part in a partition.