Sum_{k=0n} p(k) where p(k) = number of partitions of k

  • Thread starter ozymandius5
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In summary, the conversation discusses the speaker's interest in learning more about polynomials and generating functions. They mention a zipped document and suggest looking at a specific webpage for further information and possibly finding a generating function.
  • #1
ozymandius5
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Very much a beginner in maths and broadening my horizons. I have a series of polynomials that I was hoping to get some insight into, specifically where to beginning looking re. a method of creating a generating function, as well as some self similar patterns and links that explain them. Any help would be greatly appreciated.

I think the zipped .doc and .xls attachment lays out the broad strokes.
 

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  • Integer Partitions.zip
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  • #2
If you haven't done so, take a look at this page. Among other things that might interest you, a generating function is provided there.

EDIT: Considering the thread title is an exact copy of the title of the series in the link, I guess you have seen it. Dig deeper and you'll find a g.f. :)
 

1. What is "Sum_{k=0n} p(k) where p(k) = number of partitions of k"?

"Sum_{k=0n} p(k) where p(k) = number of partitions of k" is a mathematical expression that represents the sum of all possible partitions of the number n. A partition of a number is a way of breaking it down into smaller positive integers. For example, the partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.

2. Why is the sum of partitions important in mathematics?

The sum of partitions has many applications in various fields of mathematics, such as number theory, combinatorics, and algebra. It can be used to solve problems in areas such as counting, probability, and symmetry. Additionally, understanding the patterns and properties of partitions can provide insights into other mathematical concepts.

3. Is there a formula for calculating the sum of partitions?

Yes, there is a formula for calculating the sum of partitions known as the Hardy-Ramanujan formula. It states that the sum of partitions of a number n can be approximated by the expression √(n/3) * e^(π√(2n/3)). However, this formula is an approximation and becomes more accurate as n gets larger.

4. Can the sum of partitions be represented in other ways?

Yes, there are various other ways to represent the sum of partitions, such as generating functions, recurrence relations, and congruences. These representations can provide different insights and approaches to solving problems related to partitions.

5. Are there any practical applications of the sum of partitions?

Yes, the sum of partitions has practical applications in fields such as computer science and physics. In computer science, partitions can be used to optimize algorithms and data structures. In physics, partitions can be used to model the energy levels of particles in a system.

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