What are interesting integer partition problems?

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[enpeacemusic]

I love math and how incredibly broad it is

[andrewfucarino]

I thought maybe you’d talk about my favorite concept related to partitions, Landau’s function! The partition of n with the largest LCM. Really fun to find by hand

[derekhasabrain]

I came across one of your videos a few months ago, I don’t exactly remember the problem but in my mind it was very similar to blackjack so I decided to mess around with my own version of that problem. The question was essentially “how many winning states are there in blackjack” but I wanted to prove it with my limited math skills. I came across partitions and immediately felt way out of my depth. It’s fascinating

[HaramGuys]

its not taught in number theory but taught in combinatorics

[timelsen]

Also used in advanced quantum mechanics.

[spiderjerusalem]

Proof by generating function

the func corresponding to partition into odd parts:

=
= Π⁰⁰ₖ₌₁ 1/(1–x²ᵏ⁻¹)
= Π⁰⁰ₖ₌₁ (1–x²ᵏ)/[(1–x²ᵏ⁻¹)(1–x²ᵏ)]
= Π⁰⁰ₖ₌₁ (1–xᵏ)(1+xᵏ)/(1–xᵏ)
= Π⁰⁰ₖ₌₁ (1+xᵏ)
= (1+x)(1+x²)(1+x³)...(1+xⁿ)...
= numbers of partitions into distinct parts of size, each of which is counted at most once

[lior_shiboli]

Fun fact partitions are really helpful in some soduku variants especially killer soduku

[dougr.]

Au contraire, I own a very rare number theory “elementary” text that has a discussion of partitions in its last chapter.

[dr.rahulgupta]

Sir, You must have read the works of Sriniwas Ramanujan !