Simultaneous core partitions with nontrivial common divisor

Acknowledgements Part of this work was done at the Centre Interfacultaire Bernoulli (CIB), in the École Polytechnique Fédérale de Lausanne (Switzerland), during the Semester Local Representation Theory and Simple Groups. The first two authors are grateful to the CIB for their financial and logistical support. The first author also acknowledges financial support from the Engineering and Physical Sciences Research Council grant Combinatorial Representation Theory EP/M019292/1. The second author was supported by 586 PSC-TRADA-46-493 and thanks George Andrews who supported a visit to Penn State where this research began. The second author also thanks Christopher R. H. Hanusa for helpful conversations on diagrams and references, and notes that some diagrams were made using the ytab package. All of the authors thank the anonymous referee for the careful reading and detailed and helpful suggestions.Peer reviewedPostprin