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New Congruences Modulo 2, 4, and 8 for the Number of Tagged Parts Over the Partitions with Designated Summands. (arXiv:1807.01074v1 [math.NT])

来源于:arXiv
Recently, Lin introduced two new partition functions $\textup{PD}_\textup{t}(n)$ and $\textup{PDO}_\textup{t}(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $\textup{PD}_\textup{t}(n)$ and $\textup{PDO}_\textup{t}(n)$, and conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia found two new infinite families of congruences modulo 9 for $\textup{PD}_\textup{t}(n)$. In this paper, we prove the congruences modulo 8 conjectured by Lin and also find many new congruences and infinite families of congruences modulo some small powers of 2. 查看全文>>