: Distinct and Odd Partitions Puzzle : the solution

: Let's write out the distinct and odd partitions of 6 again:

 : The Distinct Partitions of 6: : 6 : 5 1 : 4 2 : 3 2 1 : The Odd Partitions of 6: : 5 1 : 3 3 : 3 1 1 1 : 1 1 1 1 1 1 1

: I claim that {6} becomes {3 3}, {5 1} becomes {5 1}, {4 2} becomes {1 1 1 1 1 1}
: and {3 2 1} becomes {3 1 1 1}, and vice versa. Do you see it yet?

: Here's the bijection, described algorithmically:

: To map from a distinct to an odd partition, we decompose each even number 'k'
: in the distinct partition into a pair of numbers k1 and k2, where k1 = k2 (i.e. we
: divide k by 2). If these numbers are still even, then we repeat the process ad
: infinitum until we are left with only odd numbers. A distinct partition that is
: also an odd partition, we simply leave as-is. Try this algorithm with {4 2}.

: To map from an odd to a distinct partition, we take the first pair of repeated
: values and add them. We continue this process moving from right to left, until
: we have a distinct set. As above, an odd partition that is also a distinct partition
: we leave as-is. Try this approach with {1 1 1 1 1 1 }.

: Try it on a more complicated partition (thanks to James Corey for correcting a mistake in this example):

 : The Distinct Partitions of 10: : 10 : 9 1 : 8 2 : 7 3 : 7 2 1 : 6 4 : 6 3 1 : 5 4 1 : 5 3 2 : 4 3 2 1 : The Odd Partitions of 10: : 9 1 : 7 3 : 7 1 1 1 : 5 5 : 5 3 1 1 : 5 1 1 1 1 1 : 3 3 3 1 : 3 3 1 1 1 1 : 3 1 1 1 1 1 1 1 : 1 1 1 1 1 1 1 1 1 1 1