A tremendous amount of research has been done in the last two decades on (s, t)-core partitions when s and t are relatively prime integers. Here we change perspective slightly and explore properties of (s, t)-core and \((\bar{s},\bar{t})\)-core partitions for s and t with a nontrivial common divisor g. We begin by recovering, using the g-core and g-quotient construction, the generating function for (s, t)-core partitions first obtained by Aukerman et al. (Discrete Math 309(9):2712–2720, 2009). Then, using a construction developed by the first two authors, we obtain a generating function for the number of \((\bar{s},\bar{t})\)-core partitions of n. Our approach allows for new results on t-cores and self-conjugate t-cores that are notg-cores and \(\bar{t}\)-cores that are not\(\bar{g}\)-cores, thus strengthening positivity results of Ono and Granville (Trans Am Soc 348:221–228, 1996), Baldwin et al. (J Algebra 297:438–452, 2006), and Kiming (J Number Theory 60:97–102, 1996). We then move to bijections between bar-core partitions and self-conjugate partitions. We give a new, short proof of a correspondence between self-conjugate t-core and \(\bar{t}\)-core partitions when t is odd and positive first due to Yang (Ramanujan J 44:197, 2019). Then, using two different lattice-path labelings, one due to Ford et al. (J Number Theory 129:858–865, 2009), the other to Bessenrodt and Olsson (J Algebra 306:3–16, 2006), we give a bijection between self-conjugate (s, t)-core and \((\bar{s},\bar{t})\)-core partitions when s and t are odd and coprime. We end this section with a bijection between self-conjugate (s, t)-core and \((\bar{s},\bar{t})\)-core partitions when s and t are odd and nontrivial g which uses the results stated above. We end the paper by noting (s, t)-core and \((\bar{s}, \bar{t})\)-core partitions inherit Ramanujan-type congruences from those of g-core and \(\bar{g}\)-core partitions.

在过去的二十年中，当s和t是相对质数整数时，已经对（s，  t）核心分区进行了大量研究。在这里，我们稍微改变视角和探索（的性质小号， 吨）-core和\（（\巴{S}，\ {酒吧吨}）\）为-core分区小号和吨与非平凡公约数克。我们首先使用g -core和g -quotient构造恢复（s，  t）核心分区首先由Aukerman等人获得。（离散数学309（9）：2712–2720，2009年）。然后，使用由前两个作者开发的结构，我们得到了一个生成函数为的数目\（（\巴{S}，\ {酒吧T】）\） -core的分区Ñ。我们的方法允许在非g -core的t -core和自共轭t -core和非\（\ bar {g} \）的\（\ bar {t} \）- cores上获得新结果。核心，从而加强了Ono和Granville的阳性结果（Trans Am Soc 348：221–228，1996），Baldwin等。（J Algebra 297：438-452，2006）和Kiming（J Number Theory 60：97-102，1996）。然后，我们转到条形核心分区和自共轭分区之间的双射。我们给出了一个新的简短证明，即当t首先由于阳数而为奇数且为正数时，自共轭t -core分区与\（\ bar {t} \）- core分区之间的对应关系（Ramanujan J 44：197，2019）。然后，使用两种不同的晶格路径标记，一种是由于福特等人的缘故。（J Number Theory 129：858–865，2009），另一本是Bessenrodt and Olsson（J Algebra 306：3–16，2006），我们给出了自共轭（s，  t）核心与\（（\ bar {s}，\ bar {t}）\）-当s和t为奇数和互质时的核心分区。我们以自共轭（s，  t）核心与\（（\ bar {s}，\ bar {t}）\）- core分区之间的双射结束本节，当s和t为奇数且非平凡g时，使用上述结果。最后，我们注意到（s，  t）核心和\（（\ bar {s}，\ bar {t}）\）-核心分区从g -core和\（\ bar继承了Ramanujan类型的一致性{g} \）-核心分区。