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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.

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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.

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