# ON THE MULTI-DIMENSIONAL PARTITIONS OF SMALL INTEGERS

Kim, Jun-Kyo

• Received : 2011.05.03
• Accepted : 2011.12.16
• Published : 2012.01.31
• 29 7

#### Abstract

For each dimension exceeds 1, determining the number of multi-dimensional partitions of a positive integer is an open question in combinatorial number theory. For n ${\leq}$ 14 and d ${\geq}$ 1 we derive a formula for the function ${\wp}_d(n)$ where ${\wp}_d(n)$ denotes the number of partitions of n arranged on a d-dimensional space. We also give an another definition of the d-dimensional partitions using the union of finite number of divisor sets of integers.

#### Keywords

partitions of integers;multidimensional partition;combinatorial number theory;additive number theory

#### References

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#### Acknowledgement

Supported by : Pusan National University