The Number of Partitions into Odd Parts Equals the Number of Partitions into Distinct Parts
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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.
Contributed by: George Beck (March 2011)
Open content licensed under CC BY-NC-SA
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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.
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