Partioning natural numbers

[quantumfireball]

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




$$\Gamma$$(M,N) as the no of ways to partition M into N whole numbers subject to condition 1)
$$\Pi$$(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.

 

[quantumfireball]

if you read carefully you will find a striking resemblance of the number theoretical problem and quantum statistics,wherin $$\Pi$$(M,N) gives the number of microstates asscociated with N harmonic oscillators having total energy MhV.
i read something similar in Zweibachs string theory.

 

[al-mahed]

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