We consider (k,j)-colored partitions, partitions in which k colors exist but at most j colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including in previously unexplored cases where k and j are not coprime, as well as some noncongruences. As a useful aside, we give the apparently new generating function for the number of partitions in the N × M box with a given number of part sizes, and extend to multiple colors a conjecture of Dousse and Kim on unimodality in overpartitions.

中文翻译：

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受限的 k 色分区，II

我们认为(ķ,j)-彩色分区，其中的分区ķ颜色存在，但最多j可以根据零件尺寸选择颜色。特别是这些概括了过度分区。推进以前的工作，我们发现了新的一致性，包括在以前未探索的情况下ķ和j不是互质的，还有一些不全等的。顺便说一句，我们给出了显然是新的生成函数，用于ñ × 米具有给定数量的零件尺寸的盒子，并扩展到多种颜色，这是 Dousse 和 Kim 关于过度分区中单峰性的猜想。