I am reading Macdonlad's book on "symmetric functions and Hall polynomials" and I have difficulty figuring out an identity which involves hook-lengths. I would like to ask for a hint. 


Let $\lambda=(\lambda\_1,\dots, \lambda\_k)$ be a partition and define $\mu\_i=\lambda\_i+k-i$, for $1\leq i\leq k$. The hook-length of a box $x\in \lambda$ in the Young diagram is denoted by $h(x)$. One can show the following identity: 
$$
\sum\_{x\in \lambda}t^{h(x)}+\sum\_{1\leq i<j\leq k}t^{\mu\_i-\mu\_j}=\sum\_{i=1}^k\sum\_{j=1}^{\mu\_i}t^j.
$$ 
From this identity, Macdonald concludes the following identity which I am not able to verify:
$$
\prod\_{x\in \lambda}\left(1-t^{h(x)}\right)\cdot \prod\_{1\leq i<j\leq k}\left(1-t^{\mu\_i-\mu\_j}\right)=\prod\_{i=1}^k\prod\_{j=1}^{\mu\_i}\left(1-t^j\right)
$$
How can I conclude the second identity from the first one?



An identity involving hook-lengths
The first identity says that
$$\{h(x) : x \in \lambda\} \cup \{\mu\_i - \mu\_j : 1 \leq i < j \leq k\} = \bigcup\_{i=1}^k \{j : 1 \leq j \leq \mu\_i\}$$
as multisets. The second identity then follows because the multisets of exponents are the same on the LHS and RHS.


