Integer Partition Calculator
Discover the fascinating world of integer partitions. Enter a positive integer to calculate the number of ways it can be partitioned.
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An integer partition of a positive integer n is a way of writing n as a sum of positive integers. Order of addends does not matter. For example, the partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.
Partition Visualization
Each dot represents a unit, visualizing the integer . The total number of partitions, , indicates the different ways these dots can be grouped to sum up to the original integer.
Integer: | Number of Partitions:
Understanding Integer Partitions
In number theory, an integer partition of a positive integer n is a way of writing n as a sum of positive integers. Two partitions that differ only in the order of their summands are considered the same partition. For example, 4 can be partitioned in 5 ways: 1+1+1+1, 1+1+2, 1+3, 2+2, 4. The number of partitions grows rapidly with n. Integer partitions are studied in combinatorics and number theory, finding applications in various fields like statistical mechanics and computer science.
- Example: For n=5, there are 7 partitions: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1.
- Formula: There is no simple closed-form formula for the partition function p(n), which gives the number of partitions of n. However, there are recursive formulas and asymptotic approximations.
- Further Reading: Wikipedia on Integer Partitions