The Number of Partitions into Odd Parts Equals the Number of Partitions into Distinct Parts
In 1748 Euler proved that for any , the number of partitions of into odd parts is the same as the number of partitions of into distinct parts. This Demonstration shows how to go from either type to the other and back.
A partition of 6, for example, is a sum like with parts , , . The four partitions of into parts all of which are odd are , , , and . The four partitions of into parts that are distinct are , , , .
To go from an odd partition to a distinct partition, first collect the parts, , so that are distinct odd numbers. Write each corresponding as a binary number, , where or , . Multiplying out all terms gives a sum of distinct parts, , because the are distinct and the are distinct for a given .
Going the other way, from a distinct partition , let be a typical part. Let be the highest power of 2 dividing . Factor as and write , where there are summands . This is a sum of odd parts. Do this for each distinct part to partition into odd parts.
These two constructions are inverses of each other, so the set of odd partitions and the set of even partitions have the same size.