sequences and series - how interpret this partition identity?

use the symbol $P(N)$ to denote the set of all partitions of a positive integer $N$ and denote by $P_k$ the number of occurrences of $k$ in the partition $P \in P(N)$, so that
$$
N = \sum kP_k

$$

by equating coefficients in the identity:
$$
\frac1{1-x} = e^{-\ln(1-x)}=e^{\sum_{k=1}^{\infty}\frac{x^k}{k}}
$$
we see that
$$
\sum_{P \in P(N)}\left( \prod_{P_k\gt 0}k^{P_k}P_k!\right)^{-1} = 1 \tag{1}
$$

Question (a) does this identity have any well-known combinatorial interpretation? (b) is there a simple direct proof of (1) which does not invoke power series?

Answer

hint: look at http://lipn.univ-paris13.fr/~duchamp/Books&more/Macdonald/%5BI._G._Macdonald%5D_Symmetric_Functions_and_Hall_Pol%28BookFi.org%29.pdf pg 24 (2.14) giving the $z_\lambda$ ; your expression is $z_\lambda/n!$ known as the inverses of the class sizes of the symmetric group $S_n$.
Example: for $S_5$ we get class sizes 24, 30, 20, 20, 15, 10, 1 adding to $5!$ or 120 (order of the group). Divide them by $5!$ and the sum gets to be 1.