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Generating functions are useful in studying the number of different types of partitions of an integer \(n .\) A partition of a positive integer is a way to write this integer as the sum of positive integers where repetition is allowed and the or- der of the integers in the sum does not matter. For exam- ple, the partitions of 5 (with no restrictions) are \(1+1+1+1+\) \(1+1,1+1+1+2,1+1+3,1+2+2,1+4,2+3,\) and \(5 .\) Exercises \(53-58\) illustrate some of these uses. Show that the coefficient \(p_{d}(n)\) of \(x^{n}\) in the formal power series expansion of \((1+x)\left(1+x^{2}\right)\left(1+x^{3}\right) \cdots\) equals the number of partitions of \(n\) into distinct parts, that is, the number of ways to write \(n\) as the sum of positive inte- gers, where the order does not matter but no repetitions are allowed.

Step by step solution

01

Understanding the Generating Function

Recognize that we need to show the coefficient of the term containing \(x^n\) in the series expansion of \((1+x)(1+x^2)(1+x^3)\cdots\) corresponds to the number of partitions of \(n\) into distinct parts.

02

Expand the Generating Function

Expand the generating function:\[(1+x)(1+x^2)(1+x^3)\cdots\]This can be seen as a product of binomials where each term \(1+x^k\) has coefficients 1 and 1 for \(x^k\).

03

Form Partitions of n

Write out the individual terms by expanding and combining different products of \(x^k\)'s to form different sums that add up to \(n\). Each combination forms a unique partition into distinct parts.

04

Identify Coefficient of x^n

The coefficient of \(x^n\) in the expanded series is the number of unique combinations of distinct parts that add up to \(n\).

05

Conclusion

Thus, the coefficient \(p_d(n)\) of \(x^n\) in the generating function \((1+x)(1+x^2)(1+x^3)\cdots\) represents the number of partitions of \(n\) into distinct parts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integer Partitions

In discrete mathematics, an integer partition is a way of writing a positive integer as a sum of other positive integers. The order of the summands does not matter, but repetitions are allowed. For example, the partitions of 4 are:

• 4
• 3 + 1
• 2 + 2
• 2 + 1 + 1
• 1 + 1 + 1 + 1

Each partition is a unique combination and represents a different way of summing up to the target integer. By understanding integer partitions, you can explore complex combinatorial properties and find solutions to related problems in number theory.

Formal Power Series

A formal power series is an infinite series, usually in the form of \[ a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \]

It is a powerful tool in algebra and combinatorics. It does not consider convergence but treats the series purely as a formal object. The coefficients of the series represent significant combinatorial information.

In the context of generating functions, consider the series \[(1 + x)(1 + x^2)(1 + x^3) \ldots \]. By expanding it, each term in the series reflects a unique way of partitioning integers into distinct parts. The coefficient of \ x^n \ in this power series reveals how many ways an integer \ n \ can be partitioned into distinct parts.

Distinct Parts in Partitions

A partition of an integer into distinct parts means that each summand must be unique and appear exactly once. For example, consider the integer 5. Its partitions into distinct parts are:

• 5
• 4 + 1
• 3 + 2

Notice that sums like '2 + 2 + 1' are not included because 2 appears more than once. Distinct part partitions are a refined way of breaking down integers, with applications in generating functions and combinatorial proofs.

The generating function \[(1 + x)(1 + x^2)(1 + x^3) \ldots \] where each binomial \(1 + x^k \) represents either taking a part k or not, encapsulates this concept efficiently, aiding in counting these partitions.