Question:

A product identity for partitions

Lily: 2 days ago

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?

Answer:
Alexander: 2 days ago

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.