Number of combinations with limited repetition

[EmileMaras]

Hello

I have the following combinatoric problem :
I want to distribute n (equivalent) atoms among M distinct objects. Each object can contain from 0 to nlim atoms. How many combination do I have for this system?

If nlim>n, this problem actually corresponds to the classical "Number of combinations with repetition". But in my case nlim<n. In fact, I am interested in the limit of (lnΩ)/n (Ω beeing the number of combination) when M and n tend toward infinity (with n=a M where a is a constant) while nlim is finite (and actually rather small)..

I found a solution for that problem using some series of summations but it will be impossible to caculate as soon as M and n become large (even for M=100, n=300 and nmax=10, it took my laptop more than one hour to solve it).
Is there a simple analytical solution to this problem?

Thank you for your help.

Emile Maras

 

[ShayanJ]

For the first atom,there is M choices...for the second,again M choices...for the third,again M choices...and so on!
So there is always M possible choices and all that we should do is to multiply the number of choices for each of the atoms which becomes $M^n$. But because the atoms are identical,we should decrease this amount by dividing it by $n!$.

 

[EmileMaras]

I guess that it is not the correct answer. Maybe I did not state my problem properly, so I will just give an exemple.
Let's say I have n=3 atoms and M=3 object. An object can contain at max nmax=2 atoms. Then the possible combinations are 111, 012, 021, 102, 120, 201, 210 (where xyz gives the number of atom in each object) which corresponds to 7 combinations.

 

[ShayanJ]

I guess that it is not the correct answer. Maybe I did not state my problem properly, so I will just give an exemple.
Let's say I have n=3 atoms and M=3 object. An object can contain at max nmax=2 atoms. Then the possible combinations are 111, 012, 021, 102, 120, 201, 210 (where xyz gives the number of atom in each object) which corresponds to 7 combinations.

Yeah,my answer is wrong.It even gives a non-integral value!
Anyway,Check here!

 

[CellCoree]

Hello Emile,

Thank you for presenting your combinatoric problem. From my understanding, you are interested in finding the number of combinations for distributing n equivalent atoms among M distinct objects, where each object can contain from 0 to a finite number nlim of atoms. The limit of (lnΩ)/n is of interest to you as M and n tend towards infinity while nlim remains finite.

Unfortunately, there is no simple analytical solution to this problem. As you have mentioned, even for relatively small values of M, n, and nlim, the calculations become extremely complex and time-consuming. In fact, this problem falls under the category of "combinatorial explosion," where the number of possible combinations increases exponentially with the number of variables.

However, there are some approaches that can help simplify this problem. One approach is to use generating functions, which can help represent the combinations in a compact form. Another approach is to use asymptotic analysis, which approximates the solution for large values of M and n. Both of these approaches may be useful in finding an approximate solution to your problem.

I hope this helps in your research. Best of luck in finding a solution to your combinatoric problem.

Sincerely,