Some New Results on the Number of Tagged Parts Over the Partitions with Designated Summands (Contemporary Mathematics, 2024)
The partitions with designated summands were introduced. In such partitions, among those parts of the same magnitude, one is tagged or designated. This work presents some new results on partitions with designated summands. The total number of partitions of n with designated summands is denoted by PD(n). PDO(n) indicates how many partitions of n there are with designated summands where every component is odd. The number of tagged parts across all partitions of n with specified summands is known as PDt(n), and the number of tagged parts across all partitions of n with designated summands in which all parts are odd is known as PDOt(n). In this study, we demonstrate a few new congruences modulo 2 and 4. We employed the dissection approach to access our recent discoveries.
Keywords: designated summands; partitions; dissections; congruence