Euler's Identity
$$
\begin{align}
p(n \mid \text{ odd parts }) = p(n \mid \text{ distinct parts})
\end{align}
$$
That is, the number of integer partitions made of odd parts is the same as the number of integer partitions made entirely from distinct parts.
Example 1
Take \(n = 6\). Then using only odd parts, we can write
$$
\begin{align}
6 &= 1 + 1 + 1 + 1 + 1 + 1 \\
6 &= 3 + 1 + 1 + 1 \\
6 &= 3 + 3 \\
6 &= 5 + 1
\end{align}
$$
Using distinct numbers, we can write
$$
\begin{align}
6 &= 1 + 2 + 3 \\
6 &= 2 + 4 \\
6 &= 5 + 1 \\
6 &= 6
\end{align}
$$
Example 2
Take \(n = 9\). Then using only odd parts, we can write
$$
\begin{align}
9 &= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 \\
9 &= 3 + 1 + 1 + 1 + 1 + 1 + 1 \\
9 &= 3 + 3 + 1 + 1 + 1 \\
9 &= 3 + 3 + 3 \\
9 &= 5 + 1 + 1 + 1 + 1 \\
9 &= 5 + 3 + 1 \\
9 &= 7 + 1 + 1 \\
9 &= 9
\end{align}
$$
Using distinct numbers, we can write
$$
\begin{align}
9 &= 4 + 3 + 2 \\
9 &= 5 + 3 + 1 \\
9 &= 6 + 2 + 1 \\
9 &= 6 + 3 \\
9 &= 5 + 4 \\
9 &= 7 + 2 \\
9 &= 8 + 1 \\
9 &= 9
\end{align}
$$
A Bijection
One way to prove the identity is to come up with a bijection between the two sets. To go from odd parts to distinct parts, then a natural thing to do is take any two copies that are equal and merge them together. This operation terminates since if we keep merging, then the number of parts will decrease at each step until every part is distinct. For example
$$
\begin{align}
9 &= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 \\
9 &= 2 + 2 + 2 + 2 + 1 \\
9 &= 4 + 4 + 1 \\
9 &= 8 + 1
\end{align}
$$
and another example
$$
\begin{align}
9 &= 3 + 1 + 1 + 1 + 1 + 1 + 1 \\
9 &= 4 + 2 + 2 + 1 \\
9 &= 6 + 2 + 1
\end{align}
$$
The inverse of this operation is going from distinct parts to odd parts. The obvious natural thing to do is to take any even number and break it into two equal parts. We stop until we there are no more even numbers.
$$
\begin{align}
9 &= 4 + 3 + 2 \\
9 &= 2 + 2 + 3 + 2 \\
9 &= 1 + 1 + 1 + 1 + 3 + 1 + 1
\end{align}
$$
References