Suppose we have a partition $$\lambda =(\lambda_1,...,\lambda_n)=(\alpha_1,...,\alpha_r|\beta_1,...,\beta_r)$$ written in the Frobenius notation.
I am trying to prove the following relation $$\sum\limits_{i=1}^n t^i(1-t^{-\lambda_i})=\sum\limits_{j=1}^r(t^{\beta_j+1}-t^{-\alpha_j}).$$
My idea was to use similar relations for hook-length, but it seems like this is a wrong way. Any hint how to start it?