# Partition theory

1. Aug 11, 2003

### MathematicalPhysicist

does p.t concern also with the multiples of a number, for example: the multiples of 4 are 2*2 and 4*1 meaning two?

p.s
i know that partition theory is concerned in the sums of a number (the partition of 4 is 5).

2. Aug 11, 2003

### marcus

Loop, could you simply just refresh our memories about ordinary ADDITION partition theory?

Like, how do you figure out how many ways there are to write the number seven as a sum?

You are jumping ahead too fast. I cannot even remember the addition part.

I think that you would call the multiplication analog of that a theory of "factorization"
like how many ways can you factorize the number 24?
and I think that the main results having to do with factorization are theorems about prime numbers and prime factorization.
It would be a separate thing from the additive business you call "partitioning".

Partitioning is interesting in its own right. Even if you allow zero as a number and even if you count 2+3 and 3+2 as two separate partitions of 5. That is, you take account of the the order. I assume you know the "binomial coefficient" written as two numbers N and k in parens
and pronounced "N choose k"

/N\
\k/

and calculated N!/(k!(N-k)!)

You say "the partition of 4 is 5". How do you calculate that?
I dont happen to know a formula. Am not altogether sure what is meant either

4, 1+3, 2+2, 1+1+2, 1+1+1+1

well that is 5 all right

1 partition into one piece
2 partition into 2 pieces
1 partition into 3 pieces
1 partition into 4 pieces
adds up to 5 in all

you happen to know a formula?

3. Aug 12, 2003

### MathematicalPhysicist

partition theory doesnt account for order and you are right about the partition of four.
there are two formulas (or at least that's what i understand from this webpage: http://www.google.co.il/search?q=cache:yyMgmLqywAgJ:www.iwu.edu/~mdancs/teaching/m389/lecture_notes/Lecture_03_24.pdf+formula+in+partition+theory&hl=iw&ie=UTF-8 [Broken] ):
1. an explicit formula given by hardy and ramanujan (which is given in the webpage above).
2. the second one is: p(n){the number of partitions}=p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-12)+p(n-15)-p(n-22)-p(n-26)+...

that's all for now...

Last edited by a moderator: May 1, 2017
4. Aug 12, 2003

### marcus

Thanks, it is an interesting topic
I went to the webpage you suggested----lecture notes
by Michael Dancs for a number theory course
http://www.iwu.edu/~mdancs/teaching/m389/
and also tried a google search myself [formula partition function]
coming up with Eric Weisstein's MathWorld

Great people like Euler and Ramanujan have worked on the
partition function---most facts about it seem hard. But one source mentioned an easy fact. Did you see this?

"The number of partitions of N into exactly m parts is the same
as the number of partitions of N into parts with maximum size m."

There are 2 partitions of 5 into 2 parts, namely (4+1, 3+2)
and also 2 partitions of 5 with maximum=2, namely (2+2+1, 2+1+1+1)

There are 4 partitions of 7 into 3 parts, namely (5+1+1, 4+2+1, 4+1+1+1, 3+3+1)
and also 4 partitions of 7 with max=3, namely (3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1)

I calculated the first 10 partition numbers to see if I could detect a pattern, but I could not see one:

P1 = 1
P2 = 2
P3 = 3
P4 = 5
P5 = 7
P6 = 11
P7 = 15
P8 = 22
P9 = 30
P10 = 42

Last edited by a moderator: May 1, 2017
5. Aug 12, 2003

### MathematicalPhysicist

there's a "semi-pattern" from p(2) to p(6) the partitions are prime numbers.

6. Dec 12, 2003

### MathematicalPhysicist

so marcus, you didnt answer my original question:"does p.t concern also with the multiples of a number"? do you know?