Euler's Identity
$$
\begin{align}
p(n \mid \text{parts in }\{N\}) = p(n \mid \text{distinct parts in }\{M\})
\end{align}
$$
for \(n \geq 1\), where \(N\) is any set of integers such that no elements of \(N\) is a power of two times an element of \(N\), and \(M\) is the set containing all elements of \(N\) together with all their multiplies of powers of two.
Note here that \(M\) is the set \(\{2^ka \mid a \in N, k \geq 0\}\).
Example 1
Take \(N = \{1\}\). Then
$$
\begin{align}
M &= \{1,2,4,8,16,32,...\}
\end{align}
$$
which is the powers of two identity.
Example 2
Take \(N = \{1,3\}\). Then \(M\) contains
$$
\begin{align}
M &= \{1,2,4,8,16,32,...\} \cup \{3, 3\cdot 2, 3 \cdot 2^2, 3\cdot 2^3, 3 \cdot 2^4,...\} \\
&= \{1,2,3,4,6,12,24,32,48...\}
\end{align}
$$
Example 3
Take \(N = \{1,3,5,7...\}\), the set of odd numbers. Then
$$
\begin{align}
M &= \{1,2,4,8,16,32...\} \cup \{3,6,12,24,48,...\} \cup \{4,8,16,...\} \cup \{5,10,20,...\} \cdots \\
&= \{1,2,3,4,5,6,7,...\}
\end{align}
$$
which is exactly Euler’s Identity