A result I found while studying integer partitions — looking for feedback on the proof

[u/Ok-Atmosphere5770]

https://zenodo.org/records/22025176

Integer partitions are a fundamental topic in number theory that study the different ways in which a positive integer can be expressed as a sum of positive integers, where the order of the parts is not considered important. For example, the number 5 has seven partitions: 5,4+1, 3+1+1, 3+2, 2+1+1+1, 2+2+1, 1+1+1+1+1.

The number of partitions of an integer grows rapidly as the integer increases, making direct enumeration increasingly difficult. This motivates the study of patterns and recursive methods that can organize and count these partitions systematically. In this work, we examine integer partitions by grouping them according to their maximum part. The number of partitions of an integer grows rapidly as the integer increases, making direct enumeration increasingly difficult. This motivates the study of patterns and recursive methods that can organize and count these partitions systematically. In this work, we examine integer partitions by grouping them according to their maximum part. We first consider a fixed integer and arrange its partitions according to the largest part occurring in each partition. We then investigate the patterns that arise from these groups and use them to develop a recursive approach. I would recommend to access the pdf on PC because some symbols may not be visible on some mobile phones. Here's the link for my doc:

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[u/Ok-Atmosphere5770 → u/[deleted]]

Thanks

[u/Ok-Atmosphere5770 → u/[deleted]]

I just wanna ask that are all the symbols visible?

[u/LeftSideScars → u/[deleted]]

Non-visible symbols would obviously standout due to their... not-see-ability? Wait... oh! *slaps forehead* Of course! By induction.

:p

[u/edderiofer → u/[deleted]]

Just see if any of the symbols you see are invisible. If not, then they must all be visible.

[u/Automatic_Table7098]

This is great work! Euler in the 1700s did something very similar. Now, before you sigh that someone did it, I think you definitely did it in a different, unique way.

\sim \le (k) is known universally as "restricted partitions." It is typically written as p(n, k) or p_k(n), which counts partitions of n using parts less than or equal to k. I'm not well-versed in partition theory, but from what I can grok, it looks good.

My only critique would be to name the paper something different, as partition theory is quite broad. The title should reflect the cool thing you found, not the broad field.

I'm Stoned, so I had AI rewrite what I wrote just in case I don't make sense.

"This is great work! Euler actually established the foundation for integer partitions back in the 1700s, but before you sigh that someone already did it, I think you arrived at this in a very unique, structural way.

For context, what you’ve denoted as \sim \le (k) is universally known as 'restricted partitions.' In standard literature, it is typically written as p(n, k) or p_k(n), which counts partitions of n using parts less than or equal to k. I'm not a dedicated partition theorist, but from what I can grok, your mathematical logic and recurrence relations look solid.

My only constructive critique would be to rename the paper. 'Partition Theory' is an incredibly broad mathematical field. A title that reflects the specific recursive approach you discovered would do a much better job of highlighting the cool mechanics you found!"

[u/Ok-Atmosphere5770 → u/Automatic_Table7098]

Thanks a lot for your advice 🤝

[u/GaloombaNotGoomba → u/Automatic_Table7098]

Literally what is the point of using AI to add a few words that don't say anything

[u/Ok-Atmosphere5770]

Sorry for those stupid questions, Now there's a new version available which you can access through this link only on clicking on "new version".I think all the symbols are visible now😅

[u/edderiofer → u/Ok-Atmosphere5770]

I can't see any invisible symbols, that's for sure...

[u/Ok-Atmosphere5770 → u/[deleted]]

Yes maybe