# A combinatorial proof of an identity of partitions (Macdonald I.5)

This is a statement from Symmetric Functions and Hall Polynomials by Macdonald: $$\sum_{x\in \lambda} (h(x)^2-c(x)^2)=|\lambda|^2$$ where $$\lambda$$ denotes a partition or a Young diagram, and for each box $$x$$ in the Young diagram $$h(x)$$ is the hook length, and $$c(x)$$ is the content.

It is easy to prove this by an induction on $$|\lambda|$$, the induction step being remove the rightmost box from last row, and notice that only the hook lengths along the corresponding row and column changes (by 1) and the contents are the same, and then calculate.

Is there a direct combinatorial proof of this statement?

Even if not, is there a intuitive way to guess the statement?

• See also mathoverflow.net/questions/312771/… (not sure if any of the proofs is combinatorial). Oct 2 '20 at 12:07
• Thanks @darijgrinberg ... both the proofs there seem to require one to know first the identity somehow.
– ArB
Oct 2 '20 at 12:21