I am following Macdonald's Symmetric Functions and Hall Polynomials, and in his example-section. He stated the following without any proof, and I'm trying to understand how he reached the expression. Any assistance will be appreciated.
A partition $\lambda = (\lambda_1, \lambda_2, \cdots, \lambda_n)$ can be written in the Frobenius notation $(\alpha_1,\cdots,\alpha_r \ \vert \ \beta_1,\cdots,\beta_r)$, where $\alpha_i = \lambda_i - i$, $\beta_i = \lambda'_i - 1, \ $ $(1 \leq i \leq r)$. $r$ is the length of the intersection of the young diagram of $\lambda$ with the diagonal $i=j$.
Essentially, in the Young Diagram of $\lambda$, if we discard the diagonal, which contains $r$ squares, counting left to right to the right of the diagonal gives us $\alpha$, and counting downwards to the left of diagonal gives us $\beta$.
Macdonald states: $$\sum_{i=1}^{n}t^i\big(1-t^{-\lambda_i}\big)=\sum_{j=1}^{r}\big(t^{\beta_j + 1}-t^{-\alpha_j}\big)$$