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IMPORTANT NOTE: This paper is much rougher than I'd usually submit, and notentirely complete, though the main theorems and proofs should not be hard tofollow. Given the ongoing strike at UK Universities it may be some time beforeI get to complete it to my satisfaction, and in the meantime people I've sharedthe preliminary draft with would like to be able to reference it. Hence I'muploading it in its current form, and will update it later.  The main new result of this paper is to count the number of (n,n+1)-corepartitions with odd parts, answering a question of Zaleski and Zeilberger withbounty a charitable contribution to the OEIS. Along the way, we prove a generaltheorem giving a recurrence for (n,n+1)-core parts whose smallest part andconsecutive part differences are restricted to lie in an arbitrary set M. Thistheorem unifies many known results about (n,n+1)-core partitions withrestrictions.  We end with discussions of extensions of the general theorem that keep trackof the largest part, number of parts, and size of the partition, and about afew cases where the same methods work on more general simultaneous cores.

Simultaneous cores with restrictions and a question of Zaleski and Zeilberger