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.
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...