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Describe at least one way to generate all the partitions of a positive integer n.

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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:

 w(n)=\frac{1}{n} \sum_{k=1}^{n} a(k) m(n-k)

which includes the addition of divisors of integer n and taking w(0)=1.0

a(k) denotes the addition of divisors of k \cdot Suppose \left(w 1^{r 1}\right)\left(w 2^{r 2}\right) \ldots\left(w t^{r}\right) are the prime factorization of k, the addition of divisors of k is given by the expression:

 \mathrm{a}(k)=\prod_{j=1}^{t} \frac{w j^{i j+1}-1}{w j-1}

The sequence w(n) can be computed by using the given formula recursively:

 \mathrm{w}(n)=\sum(-1)^{(k-1)} \mathrm{w}\left(n-k \frac{(3 k \pm 1)}{2}\right)

where, addition is calculated over all k in a way that \frac{k(3 k \pm 1)}{2} lies between the values 1 ton.

Euler's generating function given by:

\frac{1}{(1-x)\left(1-x^{2}\right)\left(1-x^{3}\right) \ldots}=1+w(1) x+w(2) x^{2}+w(3) x^{3}+\ldots  

can be induced to cover the disjoint sets into any (and all possible) sets of integers. For each preferable summand \mathrm{s} , in denominator arrange a multiple of \left(1-x^{s}\right) . For example, the generating function for the partitions of n into primes is

 \frac{1}{\left(1-x^{2}\right)\left(1-x^{3}\right)\left(1-x^{5}\right)\left(1-x^{7}\right)\left(1-x^{11}\right) \ldots}=1+0 x+1 x^{2}+1 x^{3}+1 x^{4}+2 x^{5}+2 x^{6}+3 x^{7}+\ldots

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Question ID: mtid-5-stid-8-sqid-723-qpid-608