Simultaneous core partitions with nontrivial common divisor

DOI10.1007/S11139-020-00289-4arXiv1909.11808MaRDI QIDQ6325999

James A. Sellers, Jean-Baptiste Gramain, R. Nath

Publication date: 25 September 2019

Abstract: A tremendous amount of research has been done in the last two decades on $(s,t)$-core partitions when $s$ and $t$ are positive integers with no common divisor. Here we change perspective slightly and explore properties of $(s,t)$-core and $$-core partitions for $s$ and $t$ with nontrivial common divisor $g$. We begin by revisiting work by D. Aukerman, D. Kane and L. Sze on $(s,t)$-core partitions for nontrivial $g$ before obtaining a generating function for the number of $$-core partitions of $n$ under the same conditions. Our approach, using the $g$-core, $g$-quotient and bar-analogues, allows for new results on $t$-cores and self-conjugate $t$-cores that are {it not} $g$-cores and $$-cores that are {it not} $$-cores, thus strengthening positivity results of K. Ono and A. Granville, J. Baldwin et. al., and I. Kiming. We then detail a new bijection between self-conjugate $(s,t)$-core and $$-core partitions for $s$ and $t$ odd with odd, nontrivial common divisor $g$. Here the core-quotient construction fits remarkably well with certain lattice-path labelings due to B. Ford, H. Mai, and L. Sze and C. Bessenrodt and J. Olsson. Along the way we give a new proof of a correspondence of J. Yang between self-conjugate $t$-core and $$-core partitions when $t$ is odd and positive. We end by noting $(s,t)$-core and $$-core partitions inherit Ramanujan-type congruences from those of $g$-core and $$-core partitions.

Mathematics Subject Classification ID

Combinatorial aspects of partitions of integers (05A17) Partitions; congruences and congruential restrictions (11P83)

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