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?

Alexander: 2 days ago

Yes, see 2.1.7 in my survey here ( There is also a reference there to this paper ( 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.