Euler Identity

The Euler function is defined by the formula

$$\phi(t)=\prod_{i=1}^\infty(1-t^i).$$

The coefficient of $t^k$ in function $1/\phi(t)$ equals the number of all partitions of $k$.

Exercise:Use generating functions to prove that the number of partitions of $n$ with odd parts is equal to the number of partitions of $n$ with unequal parts. $\;\Box$

Lemma 7 (Euler identity)  

$$\phi(t)=\sum_{n=-\infty}^\infty(-1)^i t^{(3n^2-n)/2}.$$

The numbers $(3n^2-n)/2$ are called pentagon numbers. Compare to numbers $n$, triangular numbers $n(n+1)/2$, square numbers $n^2$.

Exercise:Prove Euler identity by constructing a map from partitions consisting of odd number of unequal parts to partitions consisting of even number of unequal parts. $\;\Box$