Recently, Lin introduced two new partition functions \(\hbox {PD}_{\mathrm{t}}(n)\) and \(\hbox {PDO}_{\mathrm{t}}(n)\), 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 \(\hbox {PD}_{\mathrm{t}}(n)\) and \(\hbox {PDO}_{\mathrm{t}}(n)\), and conjectured some congruences modulo 8. 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.

中文翻译：

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具有指定求和数的分区上标记部分的数量的模2、4和8的新同余

最近，Lin引入了两个新的分区函数\（\ hbox {PD} _ {\ mathrm {t}}（n）\）和\（\ hbox {PDO} _ {\ mathrm {t}}（n）\），它计算具有指定加数的n个所有分区上的标记部分的总数，以及具有指定加数的n个所有分区中所有部分都是奇数的标记部分的总数。Lin还证明了\（\ hbox {PD} _ {\ mathrm {t}}（n）\）和\（\ hbox {PDO} _ {\ mathrm {t}}（n）\模3和9的等价性），并猜想出一些等式8。在本文中，我们证明了Lin猜想的等式8，并且还发现了许多新的等式和2的幂次幂的无穷大。