Written by Madhav Parthasarathy

A partition of a natural number is the way of writing these numbers as the sum of other natural numbers. The first mathematician to introduce the topic of partitions was Gottfried Wilhelm Leibniz, who is more famously known as one of the “Fathers of Calculus.” He asked J. Bernoulli about the number of “divulsions,” a tearing or pulling apart, of any natural number. Essentially, he was asking about the modern equivalent of partitions of these numbers. As Bernoulli viewed these partitions, he noticed that the partitions prime values:

P(4) = 1+1+1+1
1+1+2
2+2
3+1
4+0
;5 ways

P(5) = 1+1+1+1+1
3+2
2+2+1
4+1
2+1+1+1
3+1+1
5+0
;7 ways

P(6) = 1+1+1+1+1+1
1+1+1+1+2
1+1+3
1+1+4
1+5
6+0
2+2+2
2+1+3
2+4
3+3
2+1+1+2
;11 ways

There were some exceptions such as the partitions of 7, which is 15, and 15 is not a prime number; however, there is an infinite amount of natural numbers of which the number of partitions are a prime number. The next man who had some compelling research, Leonhard Euler, created generating functions which allowed for the product(∏) to be found, which gave the number of partitions:

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By using this approach, Sylvester brought up a new way of approaching partitions. By far, the most important contributor to partitions was Srinivasa Ramanujan. Using Modular arithmetic, Ramanujan created some restrictions to the values of the Partitions of natural numbers.
P(5n+4)=0(mod 5)
P(7n+4)=0(mod 7)
P(11n+6)=0(mod 11)

Motivated by this discovery, other mathematicians began to invest more time in studying partitions. Extensive study had been dedicated to better understanding partitions. However, as a highly complex subject, their relevance and behavior are still topics to be better understood; new research which develops in the coming years may contribute to a more comprehensive understanding of both the nature and applications of mathematical partitions.

Citations:
Andrews, George E. “Partitions.” https://www.math.psu.edu/vstein/alg/antheory/preprint/andrews/chapter.pdf

Figure 1. Boruah, Chayanika. “Partition Theory of Numbers: An interesting research area in Mathematics.” Good Morning Science, 27 May 2018

Figure 2. Bose, Raj C., and Branko Grünbaum. “Combinatorics.” Encyclopedia Britannica.