2. Describe at least one way to generate all the partitions of a positive integer n. (You can get idea from Exercise 49 in Section 5.3.)
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Partition of an integer is the representation of the positive integer n as a collection of positive integers.
Denote w(n) to be the number of ways of representing a positive integer n without considering the order of the integer. For a given integer n the number of partitions w(n) can be determined in a number of ways. Consider the given expression:
which includes the addition of divisors of integer n and taking w(0)=1.0
a(k) denotes the addition of divisors of Suppose are the prime factorization of k, the addition of divisors of k is given by the expression:
The sequence w(n) can be computed by using the given formula recursively:
where, addition is calculated over all k in a way that lies between the values 1 ton.
Euler's generating function given by:
can be induced to cover the disjoint sets into any (and all possible) sets of integers. For each preferable summand , in denominator arrange a multiple of . For example, the generating function for the partitions of n into primes is