36. Partitions
 
Home Menu Previous Next


Partitions (Basics)

Divide an integer into any number of positive integral parts. This is a partition of the number. The number of all possible partitions of a number is denoted by p(n)

NUMBERPARTITIONSNUMBER OF PARTITIONS
n  p(n)
11p(1) = 1
21+1, 2 p(2) = 2
31+1+1, 1+2, 3 p(3) = 3
41+1+1+1, 1+1+2, 1+3, 2+2, 4p(4) = 5
51+1+1+1+1, 1+1+1+2, 1+1+3, 1+4, 2+2+1, 2+3, 5p(5) = 7
6p(6) = 11
7p(7) = 15
......
20p(20) = 627
......
50p(50) = 204,226

1. As n increases, p(n) increases rapidly.
 
2. Look at the partitions of 5:
p(5): 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+4, 2+2+1, 2+3, 5

Of these 1+1+1+1+1, 1+1+3, 2+2+1 are ODD PARTS as there are odd number of parts

And 1+4, 2+3, 5 are DISTINCT PARTS as no part is repeated.

p(5) has 3 distinct parts and also, p(5) has 3 odd parts.
 
3. Euler's Theorem states: In partitions of any number, Distinct parts equal Odd parts.
 
4. Is it possible to say if p(n) is odd or even, for any n?
 
5. In the Theory of Partitions, Hardy and Ramanujan contributed a remakable theorem.

<<< Becoming Ramanujan