On Congruences for Tagged Partition Functions with Designated Summands: Partial Proofs of Lin’s Conjecture

The examination of partition functions with specified summands has uncovered intricate mathematical frameworks that expand upon traditional partition theory. Lin proposed two partition functions, 𝑃𝐷𝑂𝑡(𝑛)and 𝑃𝐷𝑂𝑡(𝑛), which enumerates the total number of tagged components across all partitions with specified summands, including those limited to odd parts. Lin established several congruences modulo small powers of 3 and proposed conjectural generalisations pertaining to larger powers of 3. This study offers partial proofs of Lin’s hypothesis for certain situations, motivated by recent advancements from Chern and Hirschhorn, as well as subsequent contributions from Arman–Singh and Mehta–Kaur, emphasising congruences modulo powers of 3 for 𝑃𝐷𝑂𝑡(𝑛). By using analytic methods grounded on generating functions and intricate manipulations of deep q-series, we enhance prior findings and reveal novel congruences, thereby refining the established divisibility characteristics of 𝑃𝐷𝑂𝑡(𝑛). These discoveries enhance the comprehension of the mathematical properties of partitions with specified summands and their tagged components, and they augment previous research on tagged partition functions and associated congruence families.