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Pre-Amble

I would like to rephrase the question

PREVIOUS INSTANCE OF THIS QUESTION ( Was closed due to an alleged duplicity ).

Considering sets of four elements we can consider sets of the following type:

  • ABCD,
  • AACD,
  • AADD,
  • AAAD,
  • AAAA.

The number of types is (of course) equal to p(4)=5, where p(n) is the PartitionsP (MMa) function.

The number of set partitions for these type of sets is:

  • ABCD: 15 = BellB[4] (MMa), where BellB is the Bell Number Function
  • AACD: 11
  • AADD: 9
  • AAAD: 7
  • AAAA: 5.

The 7 set partitions of AAAD are

  • AAAD (1)
  • AA, AD (2)
  • AAD, A (2)
  • AAA, D (2)
  • AA, A, D (3)
  • AD, A, A (3)
  • A, A, A, D (4).

I would like to know the number of set partitions of an arbitrary set, for example: ABCDEF (although, for this particular case, BellB[6] works), AAAFFF, or AAAAAA, expressed in a formula, or a (programmable) algorithm.

( The Function SetPartitions in the Combinatorica Package in Mathematica can calculate the actual partitions but is rather memory-consuming, even for moderate set sizes. )

nilo de roock
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    You talk about random sets, but you don't specify a probability distribution. I get the impression that what you actually mean is arbitrary sets? – joriki Feb 08 '23 at 12:50
  • Thank you very much, good point, I will change this in the text. – nilo de roock Feb 08 '23 at 13:00
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    Does this answer your question? – J.G. Feb 08 '23 at 13:16
  • Certainly not directly, but it might be a start, or at least a hint. I will study, and test, both question and answer first. - Thanks for your reply. - If you think this Q. should be closed as a duplicate, then where is the answer in the duplicate? How many partitions are there in a set of type AAABBBCDEFGH ? – nilo de roock Feb 08 '23 at 13:29
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    When there are only two types of objects, the number of partitions is in OEIS A054225. – aschepler Feb 08 '23 at 13:37
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    Not a duplicate of 771265. That question requires that each partition has at most one of every value, but this one allows the partitions to have any multiplicities. – aschepler Feb 08 '23 at 13:43
  • I will solve this somehow. – nilo de roock Feb 09 '23 at 10:37
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    @joriki Confusing random for arbitrary is due to the fact that my native language is Dutch. In my country they think it is a simple thing to change the language at universities from Dutch to English. Just to get more foreign students. - They haven't got a clue about how sensitive the English language is, especially in Mathematics. – nilo de roock Feb 09 '23 at 13:10

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