Combinatorics Prove that pn is equal to the number of partit

*Combinatorics*

Prove that p(n) is equal to the number of partitions of the integer 2n with no odd parts. Hint: Say n=6. What is the connection between the following partitions of 6: 6 = 3 + 2 + 1 and the partition of 2*6 = 12:12 = 6 + 4 + 2?

Solution

Let the partitions of n = a1+a2+a3+...+ak

Given that p(n) is the no of partions of 2n

=2×(a1+a2+a3+...+.ak)

=2a1+2a2+2a3+....+2ak

We know that (2×any number ) is an even number.

There fore p(n) contains the only even parts.with out any odd part.