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Partition theory

  1. Aug 11, 2003 #1
    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?

    i know that partition theory is concerned in the sums of a number (the partition of 4 is 5).
  2. jcsd
  3. Aug 11, 2003 #2


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    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"


    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?
  4. Aug 12, 2003 #3
    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=ca...df+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...
  5. Aug 12, 2003 #4


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    Thanks, it is an interesting topic
    I went to the webpage you suggested----lecture notes
    by Michael Dancs for a number theory course
    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
  6. Aug 12, 2003 #5
    there's a "semi-pattern" from p(2) to p(6) the partitions are prime numbers.
  7. Dec 12, 2003 #6
    so marcus, you didnt answer my original question:"does p.t concern also with the multiples of a number"? do you know?
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