A product identity for partitions

Question

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 1

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.

Answer 2

Let me add a few remarks to Igor Pak's nice answer. First, notice that Hoare proves something stronger than the product identity, namely the observation that every number appears the same number of times on each side of the equation. In other words:

Theorem: (Stanley 1972, Elder 1984) The total number of occurrences of an integer k among all unordered partitions of n is equal to the number of occasions that a part occurs k or more times in a partition.

The case of $k=1$ is more popular, and I've seen it stated as an exercise in several occasions. Some people call the case $k=1$ Stanley's theorem and the general case Elder's theorem. Hoare gave a bijective proof, another independent proof was given by M.S Kirdar and T.H.R. Skyrme, "On an Identity Related to Partitions and Repetitions of Parts." Canad. J. Math. 34, 194-195, 1982 (Their proof uses generating functions).

One funny curiosity is that Stanley sent the general result above to the problem section of American Mathematical Monthly in 1972, but it was rejected as "a bit on the easy side, and using only a standard argument". Notice that Hoare's bijection was presented in the AMM in 1986.

One further generalization says that the number of occurrences of an integer k among all partitions of n is equal to the number of boxes among partitions of n whose hook-type is $(j,k-j-1)$. Setting $j=0$ recovers the theorem above. This was proved in C. Bessenrodt, "On hooks of Young diagrams, Ann. of Comb., 2 (1998), pp. 103–110 and in R. Bacher, L. Manivel "Hooks and Powers of Parts in Partitions", Sem. Lothar. Combin., vol. 47, article B47d, 2001.