Let $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_{\ell(\lambda)}>0)$ be an integer partition of $n\in\mathbb{N}$; i.e., $\lambda_1+\cdots+\lambda_{\ell(\lambda)}=n$.

One may now associate $f_{\lambda}=\dim(\lambda)=\#SYT(\lambda)$ which is computed by $f_{\lambda}=\frac{n!}{H_{\lambda}}$ where $H_{\lambda}=\prod_{u\in\lambda}h_u$ is the product of the hook-lengths $h_u$ of cells $u$ in the Young diagram of $\lambda$. On the other hand, for the symmetric group $\frak{S}_n$ of permutations on $n$ letters $\{1,2,\dots,n\}$, there is the cycle index formula $g_{\lambda}=\frac{n!}{z_{\lambda}}$ counting the numbers of permutations indexed by cycle-type $\lambda$. If $\lambda$ is expressed in frequency notation as $\lambda=1^{a_1}2^{a_2}\cdots n^{a_n}$ then $z_{\lambda}=1^{a_1}2^{a_2}\cdots n^{a_n}a_1!a_2!\cdots a_n!$ as a product.

Now, consider the two data of multisets (items may be repeated) $$\mathcal{F}_n=\{f_{\lambda}: \lambda\vdash n\} \qquad \text{and} \qquad \mathcal{G}_n=\{g_{\lambda}: \lambda\vdash n\}.$$

Observe $\#\mathcal{F}_n=\#\mathcal{G}_n=p(n)$, the number of partitions of $n$.

I would like to ask whether the following is true or not:

QUESTION.For any $f_{\lambda}\in \mathcal{F}_n$, there exists $g_{\mu}\in \mathcal{G}_n$ such that the fraction $\frac{g_{\mu}}{f_{\lambda}}=\frac{H_{\lambda}}{z_{\mu}}$ is actually an integer. We insist the map $\lambda\rightarrow\mu$ to be $1$-to-$1$.

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