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.
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.
