Page 26 - Ramanujan Yatra

Ramanujan with Hardy made a revolutionary change in the field of partition theory of numbers. They invented the Circle Method which gave the first approximations of the partition of numbers beyond 200. While studying this question, Hardy and Ramanujan worked with the impressive “human calculator”. The following derivations provided by Ramanujan enabled us to understand the concept of his number theory without any ambiguity.
A partition of a positive integer ‘n’ is a non-increasing sequence of positive integers, called parts, whose sum equals n. It means the number of ways in which a given number can be expressed as a sum of positive integers. For example, p (4) = 5, i.e., there are five different ways that we can express the number 4. The partitions of the number 4 are
4, 3+1, 2+2, 2+1+1, 1+1+1+1
The number of partitions of a positive integer n is denoted by p (n). For convenience, we set p (0) =1, which means it is considered that 0 has one partition. In the definition of partitions, the order does not matter; 3+1 and 1+3 are the same partitions of 4. Here, 3+1 and 1+3 are called two different compositions of 4.
The number of partitions into odd parts of a positive integer n is equal to the number of partitions of n into distinct parts. For example, we consider the
number 5. The partitions of 5 are
5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1.
The number of a partition of 5 with odd parts will be 3, i.e., 5, 3+1+1 and 1+1+1+1+1. Similarly, the number of a partition of 5 with distinct parts is also
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Ramanujan
YATRA