Simultaneous cores with restrictions and a question of Zaleski and Zeilberger

TL;DR

This paper counts (n,n+1)-core partitions with odd parts, answering a question by Zaleski and Zeilberger, and introduces a general recurrence theorem for restricted core partitions, unifying many known results.

Contribution

It provides a new counting result for (n,n+1)-core partitions with odd parts and a general recurrence theorem for restricted core partitions, extending previous work.

Findings

Counted (n,n+1)-core partitions with odd parts.

Proved a recurrence for restricted (n,n+1)-core partitions.

Unified various known results about core partitions.

Abstract

IMPORTANT NOTE: This paper is much rougher than I'd usually submit, and not entirely complete, though the main theorems and proofs should not be hard to follow. Given the ongoing strike at UK Universities it may be some time before I get to complete it to my satisfaction, and in the meantime people I've shared the preliminary draft with would like to be able to reference it. Hence I'm uploading 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)-core partitions with odd parts, answering a question of Zaleski and Zeilberger with bounty a charitable contribution to the OEIS. Along the way, we prove a general theorem giving a recurrence for (n,n+1)-core parts whose smallest part and consecutive part differences are restricted to lie in an arbitrary set M. This theorem unifies many known results about (n,n+1)-core partitions with restrictions. We end with discussions of extensions of the general theorem that keep track of the largest part, number of parts, and size of the partition, and about a few cases where the same methods work on more general simultaneous cores.