Number Theory

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• May 19th 2010, 03:07 PM
Waikato
Number Theory
An unrestricted partition (order does not count, equality of sizes is ok) partition is called self-conjugate if it is identical with its conjugate. E. g 8 = 4 + 2 + 1 + 1.

Show that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts

(optional) express this result as an identity of generating functions.
• May 19th 2010, 05:45 PM
TheArtofSymmetry
Quote:

Originally Posted by Waikato
An unrestricted partition (order does not count, equality of sizes is ok) partition is called self-conjugate if it is identical with its conjugate. E. g 8 = 4 + 2 + 1 + 1.

Show that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts

(optional) express this result as an identity of generating functions.

A partition $\lambda$ is self-conjugate if $\lambda=\lambda'$ in terms of Ferrers diagram.

The bijection can be shown graphically. See wiki.

In your example, (4,2,1,1) corresponds to 7+1 in the link.

The generating function for this is $\prod_{\text{k=odd}}(1+x^k)=(1+x)(1+x^3)(1+x^5) \cdots$.
• May 19th 2010, 05:54 PM
Waikato
partition
Hello,

Have you shown that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts?

Thanks for your answers
• May 19th 2010, 08:51 PM
TheArtofSymmetry
Quote:

Originally Posted by Waikato
Hello,

Have you shown that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts?

Thanks for your answers

Yes. If your definition of an "unrestricted" partition is simply a partition without any further constraint being given.

See the claim in the above link:

Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.

For instance, (5, 5, 4, 3, 2) |- n, where n=19, is a self-conjugate partition. It converts into the 9+7+3=19. Read the link I provided. It will give you some explanations graphically.

The link I provided only shows the sketch of the proof. I encourage you to make a full proof on your own by using the sketch of the proof in the link.