I'm using Macdonald's Symmetric Functions and Hall Polynomials as a reference, and here's what's given:
Theorem (1.7, Page 3): Let $\lambda$ be a partition and let $m \geq \lambda_1$, $n \geq \lambda'_1$. Then the $m+n$ numbers $$\lambda_i + n - i \ \ \ \ \ \ (1 \leq i \leq n), \ \ \ \ \ \ \ \ \ \ \ \ \ n - 1 + j - \lambda'_j \ \ \ \ \ \ (1 \leq j \leq m)$$ are a permutation of $\{0,1,2, \cdots, m+n-1 \}$
After giving the proof (which is fairly simple), the text says: Let $$f_{\lambda,n}(t)=\sum_{i=1}^{n} t^{\lambda_i + n - i}$$ then the theorem is equivalent to the identity $$f_{\lambda,n}(t) + t^{m+n-1}f_{\lambda',n}(t^{-1})=(1-t^{m+n})/(1-t)$$
Can anyone explain how?
$f_{\lambda,n}(t)$ is the generating function for the set of numbers $\lambda_i + n - i$ with $1 \leq i \leq n$.
$t^{m+n-1}f_{\lambda',n}(t^{-1})$ is the generating function for the set of numbers $n - 1 + j - \lambda'_j$ with $1 \leq j \leq m$.
$(1-t^{m+n})/(1-t)$ is the generating function for the set of numbers $\{0,1,2, \cdots, m+n-1 \}$.