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Partioning natural numbers

  1. Nov 10, 2008 #1
    In How many ways can one write a natural number M as a sum of N whole numbers?
    Consider the two conditions;
    1)the numbers appearing in the sum are distinct.
    2)the numbers appearing in the sum are not necessary distinct.

    eg1:eight can be written as a sum of 6 whole numbers as shown below
    8=8+0+0+0+0+0
    8=1+1+1+1+4+0
    etc..(subject to condition 2)

    eg2:8 can be written as a sum of 4 whole numbers as shown below

    8=0+1+3+4
    etc..(subject to condition 1)

    Let me make the following notations




    [tex]\Gamma[/tex](M,N) as the no of ways to partition M into N whole numbers subject to condition 1)
    [tex]\Pi[/tex](M,N) as the no of ways to partition M into N whole numbers subject to condition 2)

    this is no homework problem i formulated this on my own.
     
  2. jcsd
  3. Nov 10, 2008 #2
    if you read carefully you will find a striking resemblance of the number theoretical problem and quantum statistics,wherin [tex]\Pi[/tex](M,N) gives the number of microstates asscociated with N harmonic oscillators having total energy MhV.
    i read something similar in Zweibachs string theory.
     
    Last edited: Nov 10, 2008
  4. Nov 11, 2008 #3
    this is related to the ramanujan's partition problem, if I'm not mistaken

    how many partitions has the number 5? (or, in how many you can displace 5 rocks)

    1+1+1+1+1
    2+1+1+1
    2+2+1
    3+1+1
    3+2
    4+1
    5

    I don't know if you are considering 5+0+0+0+0 and 0+5+0+0+0 two different partitions, for instance... if not the formula already exists, is yes you have to add the number of permutations
     
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