For part 1 use this link
In this post I will discuss a technique to deal with partitioning of integers namely the generating function.Generating functions in association to partitioning was first used by Leonhard Euler.
The whole idea of generating functions is to have a power series where the coefficient of determines the number of ways in which an event can happen.
For Example, the generating function determines the number of elements of a k-element set.This is due to the fact that the number of subsets of an k-element set is equal to and in the the generating function the coefficient of is indeed .
To find the number of generating functions in a partition we need to consider how many ones are there in the partition ,how many two's ,three's and so on.In each partition the ones can occur times;thus contributing a factor of to the generating function.Similarly can occur times thus contributing a factor that equals
( to the generating function.Continuing this logic we find that the generating function for the number of partitions of an integer is: .
We can use (We can choose any x as it is only representative in character) ,to find the summation of the geometric series to be: . The logic behind building this generating function is very important. Similar logic is needed to build generating functions for problems which require that the partitions meet certain conditions. We can use generating functions to prove certain relationships between partitions even though we cannot find a closed formula. There are very good approximations for the number of partitions of the general integer, n, however.The best approximation yet was given by Rademacher. The formula is far too advanced for this paper, however.
In how many ways can n be written as the sum of two positive integers? Representations differing in only the order are considered to be the same.
Find the number of solutions to the inequality a_1 + a_2 + a_3 + a_4 \le 68 where the are non-negative integers.
Post the answers in the comments section.In the next post of this series I will prove a few properties of partitioning using generating functions.Good-Day!!