I want to find a combinatorial formula that given a certain number of integers, I can find the number of all possible groupings of these integers (such that all values belong to a single group)
Say I have 3 integers, 1, 2, 3 There would be 5 groupings:
1 2 3
1|2|3|
1 2|3
1|2 3
2|1 3
I have calculated these computationally for N = 3 to 11, but I am trying to theoretically assertain. These values are: (I believe they are correct)
num_integers num_groupings
3 5
4 15
5 52
6 203
7 877
8 4140
9 21147
10 115975
11 678570
The reason for doing this is to find the total number of partitionings of a complete graph.
Any advice, or references would be appreciated
What you are looking for is Set Partitons. The counts that you are looking for are Bell numbers, see the wikipedia article.
This is called the Bell number. When you have integer sequences you don't know about look here - OEIS.
来源：https://stackoverflow.com/questions/3430599/number-of-all-possible-groupings-of-a-set-of-values