How Many Ways to Partition N Balls into Subgroups with a Maximum of k Balls?

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SUMMARY

The discussion focuses on partitioning N balls into subgroups with a maximum of k balls per subgroup. Specific sequences from the OEIS (Online Encyclopedia of Integer Sequences) are provided for various values of k, including k = 0 (A000007), k = 1 (A000012), k = 2 (A008619), and up to k = 7 (A008636). Each sequence corresponds to the number of ways to achieve the partitioning under the specified constraints. The entire triangle of sequences can be found at A008284.

PREREQUISITES
  • Understanding of combinatorial mathematics
  • Familiarity with integer sequences
  • Knowledge of the Online Encyclopedia of Integer Sequences (OEIS)
  • Basic concepts of partition theory
NEXT STEPS
  • Research the properties of combinatorial partitions
  • Explore the OEIS entries for k = 0 to k = 7
  • Study the implications of partitioning in combinatorial optimization
  • Learn about advanced partition theory techniques
USEFUL FOR

Mathematicians, combinatorial theorists, and students studying partition theory or integer sequences will benefit from this discussion.

guyov1
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Hello.
I am trying to find out the number of options to divide N balls to subgroups in a way that each subgroup does not contain more than k balls.

Any idea?
Thanks!
 
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k = 0: http://www.research.att.com/~njas/sequences/A000007
k = 1: http://www.research.att.com/~njas/sequences/A000012
k = 2: http://www.research.att.com/~njas/sequences/A008619
k = 3: http://www.research.att.com/~njas/sequences/A001399
k = 4: http://www.research.att.com/~njas/sequences/A001400
k = 5: http://www.research.att.com/~njas/sequences/A001401
k = 6: http://www.research.att.com/~njas/sequences/A001402
k = 7: http://www.research.att.com/~njas/sequences/A008636

http://www.research.att.com/~njas/sequences/A008284 is the whole triangle.
 
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