Simultaneous core partitions with nontrivial common divisor

Abstract

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.