[Discrete Mathematics] Integer Partitions

We talk about the number of ways to partition an integer.

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We're going to use generating functions to figure out how many ways we can partition an integer. Then, we'll look at a cool proof.

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[meetverma]

The best teacher on yt rn
Thanks a lot for these lectures ❤️🙏🏻

[ajaxbjax]

Your last formula on the # of partitions with no summands divisible by 3 is also true if you rephrase it to allow the summand to be divisible by 3 but not the quantity (multiplier). For example: one 2 and two 2's are allowed, but not three 2's. Likewise, one 3 and two 3's are allowed, but not three 3's or six 3's etc. The generating function becomes (1+x+x^2 + 0 + x^4 +x^5 +0 +x^7+ ...) * (1+x^2 x^4 +0 +x^8 +x^10 +0 +x^14+ ...) *(1+x^3 x^6 +0 +x^12+x^15 +0+x^21+ ...) which is the same as your (1-x^3)/(1-x) * (1-x^6)/(1-x^2) * (1-x^9)/(1-x^3)...

[WOLF]

Good morning, Do you have a video on set partitions?

[envalemdor]

In your last example, could you explain why the generating function for 2s is
(1 - x^6) / (1 - x^2) ? I thought the formula is (1- x^n+1) / (1 - x)
since n = 4, shouldn't it be (1 - x^5) / (1 - x^2) ?

Thank you for the great tutorials!

[MarvinOGarza]

Hello, great tutorial!! Thanks for sharing. Could you please tell us what's the name of the book you were using for this, the one you took out the problem from. Thanks!

[rlira]

bruh, where's the composition video?

this is great btw.

[syedmdabid]

All the possible division of a positive integer n is called partition number of that integer. But we could find only p division, how could we find? As 3 division of 5 is(1+1+3) and (1+ 2+2) is 2, but partition of 5 is 7.

[loftkey]

Thank you so much!!! 1, 000 x better than my teacher!!!

[harpuneetkalsi]

can we use bell numbers for partitioning??

[muralidharrao]

For the last problem, while considering the summand not divisible by 3, by considering x^3 in the 1s, arent you considering a summand divisible by 3?

[obzen]

in the last question was it has to go to infinity 7 is not divisible by 3, 8 is not divisible 3, 9 is divisible so it is not included and so on?

[flight]

7:40 can't we have x2 as b > and = 2

[yusufhrj]

I'm still a kid, and i think i have to learn this

[AltuğBeyhan]

what does [x^n] mean? x^n 's coefficients?

[unremove]

Is there a mistake at 11m:13sec? Why isn't the equation x^n = (1/1-x)( 1/1-x^4)(1/1-x^12) - - similiar to what was written at 7:45 where x^n = x^n = (1/1-x)( 1/1-x^2)(1/1-x^4) ?