For a partition $\lambda=(\lambda\_1\ge \lambda\_2\ge \dots)$, let 
$m\_\lambda=\prod\_i (\lambda\_i-\lambda\_{i+1})!$ be the product of factorials of consecutive differences and let $v\_\lambda=\prod\_{i | \lambda\_i\neq 0} \lambda\_i$ be the product of its nonzero terms. Then the following identity follows from representation theory of $S\_n.$ $$\prod\_\lambda m\_\lambda=\prod\_\lambda v\_\lambda,$$ where both products are over partitions of an integer $n$. In fact for the examples I checked, it seems that if you write each side as a double product, the collection of terms on both sides is the same. However I can't see a combinatorial explanation for this fact. Does anyone know of one?



A product identity for partitions
Yes, see 2.1.7 in my survey [here](http://www.math.ucla.edu/~pak/papers/psurvey.pdf). There is also a reference there to [this paper](http://www.jstor.org/openurl?volume=93&date=1986&spage=475&issn=00029890&issue=6) by A.H.M. Hoare ("An Involution of Blocks in the Partitions of $n$", *Amer. Math. Monthly* **93**, 475–476, 1986), which gives a bijective proof. 


