I am considering to distribute N-kinds of different parts among M-different countries and I wan to know the "most probable" pattern of distribution. My question is in fact ambiguous, because I am not very sure how I can distinguish types or patterns.

Let me give an example. If I were to distribute 3 kinds of parts to 3 countries, the set of all distribution is given by a set

{aaa, aab, aac, aba, abb, abc aca, acb, acc, baa, bab, bac, bba, bbb, bbc, bca, bcb, bcc, caa, cab, cac, cba, cbb, cbc, cca, ccb, ccc}.

The number of elements is of course 33 = 27. I may distinguish three types of patterns:

(1) One country receives all parts:

aaa, bbb, ccc 3 cases

(2) One country receives 2 parts and another country receives 1 part:

aab, aac, aba, abb, aca, acc, baa, bab, bba, bbc, bcb, caa, cac, cbb, cbc, cca, ccb 17 cases

(3) Each county rceives one part respectively:

abc, acb, bac, bca, cab, cba 6 cases

These types may correspond to a partition of integer 3 with the condition that (a) number of summands must not exceed 3 (in general M). In fact, 3 have three partitions:

3, 2+1, 1+1+1

In the above case of 3×3, the number of types was the number of partitions of 3 (which is often noted p(n)). But I have to consider the case when M is smaller than N.

If I am right, the number of "different types" of distributions is the number of partitions of N with the number of summands less than M+1. Let us denote it as

p*(N, M) = p( N | the number of summands must not exceed M. )

N.B. * is added in order to avoid confusion with p(N, M), wwhich is the number of partitions with summands smaller than M+1.

Now, my question is the following:

Which type (a partition among p*(N, M)) has the greatest number of distributions?

Are there any results already known? If so, would you kindly teach me a paper or a book that explains the results and how to approach to the question?

A typical case that I want to know is N = 100, M = 10. In this simple case, is it most probable that each country receives 10 parts? But, I am also interested to cases when M and N are small, for example when M and N is less than 10.

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