# Number Theory

• May 19th 2010, 02:07 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, 04:45 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 $\displaystyle \lambda$ is self-conjugate if $\displaystyle \lambda=\lambda'$ in terms of Ferrers diagram.

The bijection can be shown graphically. See wiki.

The generating function for this is $\displaystyle \prod_{\text{k=odd}}(1+x^k)=(1+x)(1+x^3)(1+x^5) \cdots$.
• May 19th 2010, 04:54 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?

• May 19th 2010, 07:51 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?