Talk:Partition function (number theory)

This illustrates the parity of the p(k,n) function. There is evidently a "pattern" for each value of floor(n/k). It seems than when floor(n/k)=3, p(k,n)=floor(n/2)-k+2.

This is in Talk, as I have as of yet found no proof of any of this.

http://www.geocities.com/n8chz/nt.txt

Obsolete formula
There needs to be a link from Partition function (number theory), which has a totally obsolete formula, to the sample program section of UBASIC, which has the modern 1939/1956 formula. I have added the key references, but may people would not have access. If and when I have time to add the correct formula then I will add the link myself. Until then I will let you choose. AGB. 13 May 2005


 * What formula is "obsolete"? In math, formulas usually don't become obsolete. I don't think that an link to UBASIC is desirable. linas 23:03, 24 Jun 2005 (UTC)

Mahlburg etal
I updated the section on congruences, but we can probably add a touch more to it. Erica Klarreich penned a very nice introduction to partition numbers, rank, and crank in her report on recent work by University of Wisconsin, Madison PhD candidate Karl Mahlburg's that links crank with partition number congruences. Ken Ono is quoted in the article as saying that Mahlburg's work has "effectively written the final chapter on Ramanujan congruences." User:mhamrick forgot to sign on 24 June 2005


 * Do you have references, prefereably online pdfs/web links? linas 23:03, 24 Jun 2005 (UTC)

Merge
The two pages that are being proposed for merging are really two different pages on the same thing. Maybe the partition function should go under an advanced section of integer partition. -Nova666

Sounds good. Timothy Clemans 19:00, 8 April 2006 (UTC)

Yes, without a doubt. You can hardly talk about partitions without talking about the partition function anyway, and the two articles are quite similar. Holomorph 11:00, 11 April 2006 (UTC)

I would go for Partition (number theory) for the merged article --Henrygb 15:38, 4 May 2006 (UTC)

Link with Partition (number theory)
I think people should merge/combine this page with the other page "Partition (number theory)" together. — Preceding unsigned comment added by Kermatoni (talk • contribs) 00:14, 20 February 2020 (UTC)
 * The article on partitions is written in summary style, meaning that many of its subtopics are condensed into short sections with a link to a longer article on those subtopics. If we merged each of those subtopics into the main article, it would become huge, much longer than the <40kb recommended sizes for articles on Wikipedia. So that would clearly not be a good thing to do. Is there some reason that you think that the partition function subtopic is more worthy of re-expansion than the other separated subtopics, Young diagram, Gaussian binomial coefficient, Durfee square, and Young's lattice, the sub-sub-topics that have similarly been separated from this one (pentagonal number theorem and Ramanujan's congruences), or the other partition-related topics listed in the see-also section of Partition (number theory) that are not even properly incorporated into the article? —David Eppstein (talk) 00:51, 20 February 2020 (UTC)

Recurrence relations
Formula in source (where $$\mathbb{P}$$ means the positive integers):

$$\sum_{k\in\mathbb{P}} (-1)^{k-1} (p(n-k(3k-1)/2) + p(n-k(3k+1)/2))$$

Our formula:

$$\sum_{k\in\mathbb{Z}\setminus\{0\}} (-1)^{k+1} p(n-k(3k-1)/2)$$

Looks different, but it's actually the same formula, just in a more compact syntax. We can omit the additional term $$p(n-k(3k+1)/2)$$ because our sum ranges over all integers except $$0$$ (not just the positive integers) and $$k(3k+1) = -k(-3k-1) = (-k)(3(-k)-1)$$. -- Chrisahn (talk) 22:03, 22 October 2020 (UTC)

Ah, makes sense now. Looked at it for a while and wasn't able to make out the significance of the negative inputs, thanks for explaining! LaplaceFox (talk) 23:43, 2 January 2023 (UTC)

Rademacher's formula and closed-form expression
Should Rademacher's formula be considered not closed-form? Tewuzij (talk) 14:29, 23 May 2023 (UTC)
 * It's not closed-form, it contains an infinite sum. — Chrisahn (talk) 14:40, 23 May 2023 (UTC)
 * But it's not an approximation either – it converges to the correct value. Maybe we should move it from the "Approximation formulas" section to a new section? — Chrisahn (talk) 14:59, 23 May 2023 (UTC)

Extension to fractional values
Is p(n), the partition function, known for non-integral values? What would p(1/2), p(3/2), p(5/2), etc. evaluate to? Kwékwlos (talk) 00:50, 25 June 2023 (UTC)


 * This seems to be a question that you ask a lot. Wikipedia article talk pages are not proper venues for general discussion, nor for conducting original research, nor for unfounded speculations related to the article subject.  So please don't.  --JBL (talk) 22:26, 25 June 2023 (UTC)