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Say we have an urn containing 3N orbs of three different colors: red, green and blue. N of them of each color.

The orbs are Arranged necessarily in sets of three, we'll call them troikas. there are 10 types of troikas according to the combination of their constituent colors: rrr, ggg, bbb, rrg, rrb, rgb, ggr, ggb, bbr, bbg

The troikas are to be arranged in one of the $\pi(N)$ partitions of N.

I would like to have a way to count all the ways the orbs can be arranged given an specific partition and a N.

Example: Consider N=4

The partitions of 4 are:

  • $\lambda_1$=[4, 0, 0, 0] 4 sets of 1 troika
  • $\lambda_2$=[2, 1, 0, 0] 2 sets of 1 troika and 1 set of 2 troika
  • $\lambda_3$=[1, 0, 1, 0] 1 set of 1 troika and 1 set of 3 troikas
  • $\lambda_4$=[0, 2, 0, 0] 2 sets of 2 troikas
  • $\lambda_5$=[0, 0, 0, 1] 1 set of 4 troikas

Given the partition $\lambda_1$ there are $\Omega(\lambda_1)$ = 25 ways to arrange the troikas

they would be of the kind: (rrr, ggg, bbb, rgb), (rrr,ggg,bbr,gbb) and so on.

for $\lambda_3$ we have 10 of them: (rrr, rggggbbbb), (rrb, rrggggbbb) and so on.

I used Marko Riedel response in Partitioning a multiset into multisets of fixed sizes to find a generating function in terms of bell polynomial and have implemented the solution in wolfram mathematica, but I'm struggling to find answers for large Ns.]

Do you think this is the kind of problem I could get an easy to manipulate formula in terms of multinational coefficients and so on?

edit 1: just to clarify, once the troikas are put into partitions, their order is irrelevant: rggggbbbb = rgggbbbg ; rgb = bgr. You can imagine that the troikas merge together, in chunks with multiple of 3 orbs in which the order does not matter.

also (rrr,bbb,ggg)=(rrr,ggg,bbb)

for N = 10, we have the partitions followed by the number of arrangements:

[10, 0, 0, 0, 0, 0, 0, 0, 0, 0] 911    
[8, 1, 0, 0, 0, 0, 0, 0, 0, 0] 7480    
[7, 0, 1, 0, 0, 0, 0, 0, 0, 0] 6397   
[6, 2, 0, 0, 0, 0, 0, 0, 0, 0] 23836    
[6, 0, 0, 1, 0, 0, 0, 0, 0, 0] 3904    
[5, 1, 1, 0, 0, 0, 0, 0, 0, 0] 33016    
[5, 0, 0, 0, 1, 0, 0, 0, 0, 0] 1849    
[4, 3, 0, 0, 0, 0, 0, 0, 0, 0] 34336    
[4, 1, 0, 1, 0, 0, 0, 0, 0, 0] 16054    
[4, 0, 2, 0, 0, 0, 0, 0, 0, 0] 10612    
[4, 0, 0, 0, 0, 1, 0, 0, 0, 0] 706    
[3, 2, 1, 0, 0, 0, 0, 0, 0, 0] 52600    
[3, 1, 0, 0, 1, 0, 0, 0, 0, 0] 5752    
[3, 0, 1, 1, 0, 0, 0, 0, 0, 0] 8950    
[3, 0, 0, 0, 0, 0, 1, 0, 0, 0] 220    
[2, 4, 0, 0, 0, 0, 0, 0, 0, 0] 19495    
[2, 2, 0, 1, 0, 0, 0, 0, 0, 0] 17644   
[2, 1, 2, 0, 0, 0, 0, 0, 0, 0] 22723    
[2, 1, 0, 0, 0, 1, 0, 0, 0, 0] 1525    
[2, 0, 1, 0, 1, 0, 0, 0, 0, 0] 2680    
[2, 0, 0, 2, 0, 0, 0, 0, 0, 0] 1597   
[2, 0, 0, 0, 0, 0, 0, 1, 0, 0] 55    
[1, 3, 1, 0, 0, 0, 0, 0, 0, 0] 22705    
[1, 2, 0, 0, 1, 0, 0, 0, 0, 0] 3715    
[1, 1, 1, 1, 0, 0, 0, 0, 0, 0] 11200    
[1, 1, 0, 0, 0, 0, 1, 0, 0, 0] 280    
[1, 0, 3, 0, 0, 0, 0, 0, 0, 0] 2506    
[1, 0, 1, 0, 0, 1, 0, 0, 0, 0] 535    
[1, 0, 0, 1, 1, 0, 0, 0, 0, 0] 715    
[1, 0, 0, 0, 0, 0, 0, 0, 1, 0] 10    
[0, 5, 0, 0, 0, 0, 0, 0, 0, 0] 2059    
[0, 3, 0, 1, 0, 0, 0, 0, 0, 0] 3040   
[0, 2, 2, 0, 0, 0, 0, 0, 0, 0] 5695  
[0, 2, 0, 0, 0, 1, 0, 0, 0, 0] 397    
[0, 1, 1, 0, 1, 0, 0, 0, 0, 0] 1330   
[0, 1, 0, 2, 0, 0, 0, 0, 0, 0] 796    
[0, 1, 0, 0, 0, 0, 0, 1, 0, 0] 28    
[0, 0, 2, 1, 0, 0, 0, 0, 0, 0] 1030    
[0, 0, 1, 0, 0, 0, 1, 0, 0, 0] 55    
[0, 0, 0, 1, 0, 1, 0, 0, 0, 0] 82    
[0, 0, 0, 0, 2, 0, 0, 0, 0, 0] 46   
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 1
JMP
  • 22,637
Antonio Bernardo
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  • I noticed that for the first partition of any N (the partition of the kind [N,0,0,...,0]) the number of arrangements follows https://oeis.org/A254233, starting in N=0. – Antonio Bernardo Feb 14 '23 at 16:35
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    A couple of things are unclear. a) Two partitions are called $\lambda_2$ – presumably the first one is supposed to be $\lambda_1$? b) The examples you give for $\lambda_2$ seem to be examples for $\lambda_3=[1,0,1,0]$? c) You write "once the troikas are put into partitions", but the example you give, "rggggbbbb = rgggbbbg", looks as if you actually mean that all balls in all the troikas in a given part of the partition are put together and it no longer matters how they were previously arranged in troikas? d) You describe a process of "drawing", but there's actually no random element, right? – joriki Feb 14 '23 at 16:36
  • @joriki Yes, your are right in all instances, I corrected the text, tank you – Antonio Bernardo Feb 14 '23 at 16:39
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    I get $\Omega(\lambda_1)=\binom{10}{4}=210$. What are the $25$? – JMP Feb 14 '23 at 17:22
  • @JMP You are over counting by allowing arrangements that do not conserve the colors of the balls, in the sense that there are exactly 4 reds, 4 greens and 4 blues. – Antonio Bernardo Feb 14 '23 at 17:26
  • This is just a special case of the question you linked to that Marko Riedel answered, right? You want to select multisets of certain sizes from a given multiset. Are you hoping that the answer will be simpler than in that case because of the special values (same number of each item, multiset sizes always a multiple of $3$)? – joriki Feb 14 '23 at 17:36
  • @joriki Exactly. I found Marko Riedel solution to be intractable numerically. I don't know if this is because of my lack of knowledge in the subject or by the nature of the problem. As my problem is a special case I hoped it could be simplified. – Antonio Bernardo Feb 14 '23 at 17:39
  • I think it's by the nature of the problem. Mixing permutations along different characteristics (here colour, order within groups and order of the groups) usually leads to intractable problems. I'm pessimistic about the special values making any difference in that regard. But I could certainly be wrong. – joriki Feb 14 '23 at 17:41
  • I am working in this problem for 3 month now, and I don't even know the kind of answer I can hope to get. – Antonio Bernardo Feb 14 '23 at 17:43

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