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I am trying to prove a conjecture which involves the SetParitions[n] function (which requires the Combinatorica package). This function returns a list of all the set partitions of n. I'd like to take the max from each "block" in each set partition, add n+1, and then multiple the results together.

For example, if n=3, we have

SetPartitions[3]={{{1,2,3}},{{1},{2,3}},{{1,2},{3}},{{1,3},{2}},{{1},{2},{3}}}

The first set partition is {1,2,3} and only has one block. So its max is 3 and I would like to return (3+4)=7. The next partition is {1}{2,3}, so the two max values are 1 and 3. This should return (1+4)*(3+4)=5*7=35. The following parition is {1,2}{3}, which has max values 2 and 3, yielding (2+4)*(3+4)=6*7=42.

I really appreciate any assistance.

Edit: I also need the size of each block. For example, if I use {1,2}{3}, I need to use the fact that the block {1,2} is of size 2 and the block {3} is of size 1.

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    $\begingroup$ Does Times @@@ (Map[Max, SetPartitions[n], {2}] + n + 1) do what you want? $\endgroup$
    – J. M.'s lack of A.I. ♦
    May 11, 2016 at 0:50
  • $\begingroup$ Yes! Thank you! $\endgroup$
    – C Sel
    May 11, 2016 at 1:04
  • $\begingroup$ With respect to the edit, try Map[Length, SetPartitions[n], {2}]. If these suit your needs, you can write an answer to your own question. $\endgroup$
    – J. M.'s lack of A.I. ♦
    May 11, 2016 at 1:05

1 Answer 1

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I needed the following two functions:

Times @@@ (Map[Max, SetPartitions[n], {2}] + n + 1)

Map[Length, SetPartitions[n], {2}]
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