Talk:Integer partition/Archive 1

Archive 1

Untitled

This page comes from the merger of two previous articles on 16:03, 4 May 2006 (UTC). Their talk pages are Talk:Integer partition and Talk:Partition function (number theory).

Q

Hi, does anybody know an asymptotic formula for the number of partitions of k into AT MOST n parts? Somewhere, it is denoted by par(k,n). Franp9am 21:28, 9 March 2007 (UTC)

Generating partitions

This article, without ever really being clear about it, is mostly about finding out the number of partitions. That can be at most half the article. What about finding the partitions? There's no closed formula according to my lecturers. There are recursive methods, though, and one should at least be given. I have constructed one particularly inelegant approach in Haskell, and I don't really want to quote it here out of embarrassment (it is very verbose, quite stupidly for haskell). I hope someone else will offer up something. 81.23.56.9 (talk) 23:27, 24 November 2008 (UTC)

"Someone else" = Knuth. Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions, section 7.2.1.4, "Generating all partitions". Probably a version of this is available from Knuth's web site. —David Eppstein (talk) 00:05, 25 November 2008 (UTC)

It may be easier to understand your question, if you could explain what you are trying to do in more detail. I'm just a dabbler in mathematics and Wikipedia won't let me post any original material, but one of my reinventions of the wheel was the discovery of partitions. So, I may be able to provide some insights if you want to pose your problem to me directly (johnnleach@att.net). Mathematicians and computer scientists don't always think of solutions in the same perspective. I could easily write an algorithm to generate partitions, but the mathematical perspective would be to write a formula that would crank them out without testing each potential case. This is why computer scientists have little trouble generating prime numbers, but mathematicians are still searching for a prime number function. I'm a mathematician at heart, but I also recognize that sometimes you merely have to change your perspective. The acceptance of fractals was a breakthrough in changing perspectives. Still, I enjoy trying to make everything fit from a mathematician's perspective. Sometimes computer scientists forget that an algorithm may be only approximating the correct answer and has limitations, so there is always room for revisiting old problems. Perspectives are also different for engineers. Engineers tend to be interested in only the final answer, so when they include formulas in their documents, they often omit any terms that cancel. As a math minded person, I like to use formulas that describe what is going on. Therefore, I often include terms, which point out that certain things may be counterbalanced by other responses, but should that counterbalance be broken, don't forget about these other factors. I don't know what you are looking for and I may or may not be able to help you. One discovery that I've made about partitions is that I can calculate the number of partitions that contain no repeated values. If I dig out my old notes, I'm may have a few more properties to share. I find that integer partitions is a beautiful pattern. It is a pattern that could be thought of as an abstract fractal, because you can look at it in so many ways and you keep finding the same pattern within itself no matter how you twist your perspective. Unfortunately, I've found very few applications in the real world. The only one that I could come up with was something like determining the probability of finding eggs on an Easter egg hunt.--JNLII (talk) 21:09, 16 December 2008 (UTC) Another potential application, which I never explored would be the study of explosions. By measuring the size of fragments and the number of them, you are partitioning a mass. It may be possible to compare these observed partitioning results to the randomized case (integer partitioning) to describe certain properties of the mass. How strong is the glue that holds it together or did the force of explosion originate from the center?--JNLII (talk) 17:10, 17 December 2008 (UTC)

I also find it interesting that most of the information that I find on the topic fails to mention the overlap in distribution tables between partitioning integers and partitioning infinity.--208.4.152.131 (talk) 23:06, 16 December 2008 (UTC) Probably because the pattern is fractal in nature, it is also easy to determine the number of partitions with a given number of fragments or the total number of partitions that contain a specific value or set of values. The best way to describe partitions is with simplier partitions. It can not be described in algebraic terms.--JNLII (talk) 17:46, 18 December 2008 (UTC)

There are so many properties and repeating patterns within Leibniz's distribution table, that I should probably look into publishing a list of these properties. Since I would need help in reseaching those who made these discoveries before me and help in the process for submitting the article to a journal, I would welcome anyone interested in coauthoring the article with me. If this works out, I would be interested in doing the same joint project on more complex partition distributions that I've worked on involving the partitioning of different objects. For example, integer partitioning could be thought of as the number of ways that you can distribute a collection of one type of object into a crowd, say pencils. You don't care who gets a pencil, merely the number of ways that the collection can be broken up. To make the problem more complex, I started looking into the distribution of multiple collections. For example, the number of ways to distribute pencils and magnets into a crowd. Contact me if anyone is interested (johnnleach@att.net).--JNLII (talk) 19:34, 22 December 2008 (UTC)

I've also looked at the partitioning of a collection of unique objects. This has led me to discover a probability distribution that is new to me, so I don't know what to call it. This distribution however is a little flatter than the t-distribution and can be used to tell me the probability of getting a sequence in a correct order, relative to order and not distance. In other words, given a collection (A,B,C,D) to partition, what is the probability of getting a sequence with A before B? I was motivated to investigate this when studying organic chemistry and I wanted to figure out how the instructor should be grading problems that require placing things into a proper sequence. I found that most instructors don't grade these problems correctly, so I recommend keeping to problems comparing only two things at once. I also discovered that if I know the first and last values of the sequence, I have a better chance of getting a higher score if I randomly guess at the rest of the sequence, than if I know two consecutive values and try to guess the rest. The chance of getting the correct sequence remains the same, but the chance of getting a better sequence homology is different. As more and more values are known, the rule for best homology score changes, such that the extremes are not necessarily the best known values to have before guessing. I'm guessing that this will have some application in bioinformatics and understanding how DNA or amino acid sequences may naturally share homologies, even when no selective pressures apply.--JNLII (talk) 16:18, 30 December 2008 (UTC)