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).
marcus
Aug11-03, 09:19 PM
Originally posted by loop quantum gravity
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).
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?
loop quantum gravity
Aug12-03, 02:41 AM
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 ):
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...
marcus
Aug12-03, 10:01 AM
Originally posted by loop quantum gravity
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 ):
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...
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: