# A conjecture on partitions

Let $$p>q$$ be two prime numbers. Let $$\lambda$$ be a partition whose $$p$$-core is $$\lambda_p$$ and $$q$$-core is $$\lambda_q$$. Assume that $$|\lambda|>|\lambda_p|+|\lambda_q|$$. Is it true that there always exists some partition $$\mu\neq \lambda$$ with $$|\mu|=|\lambda|$$, such that $$\mu$$ also has $$p$$-core $$\lambda_p$$ and $$q$$-core $$\lambda_q$$?

I have verified this conjecture for partitions with small sizes. But I can't prove it in general. Any hint will be appreciated.

## 1 Answer

The answer is yes, and it follows from the results in my paper "A generalisation of core partitions", J. Combin. Theory 127. In that paper I define a class of partitions called $$[p:q]$$-cores. One characterisation of these partitions (Corollary 5.2 in the paper) is that $$\lambda$$ is a $$[p:q]$$-core if and only if $$\lambda$$ is the unique partition with size $$|\lambda|$$, $$p$$-core $$\lambda_p$$ and $$q$$-core $$\lambda_q$$ (in other words, if and only if there is no partition $$\mu$$ as asked for in the question). But in this case Lemma 4.8 of the paper says that the $$q$$-core of $$\lambda_p$$ coincides with the $$p$$-core of $$\lambda_q$$; call this partition $$\lambda_{pq}$$. And now Proposition 5.1 in the paper says that $$|\lambda|=|\lambda_p|+|\lambda_q|-|\lambda_{pq}|$$, contrary to hypothesis.