partition

of an integer

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July 1, 2026

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In mathematics, a partition of a positive integer n is a way to express n as a sum of positive integers. These integers are called the parts of the partition. For example, the number 7 can be partitioned in several ways, such as 5 + 1 + 1, 4 + 2 + 1, 3 + 3 + 1, and 3 + 2 + 2. The order of the parts does not matter, so 5 + 1 + 1 is considered the same partition as 1 + 5 + 1.12

The study of partitions is a significant area within combinatorics and number theory. Tools like Ferrers diagrams can be used to visualize and analyze partitions, leading to various theorems about their properties. For instance, the number of partitions of n into k parts is equal to the number of partitions of n where the largest part is k.13

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combinatorics
Partitions
A partition of a positive integer n is a representation of n as a sum of positive integers n = x₁ + x₂ +⋯+ x_k, x_i ≥ 1, i = 1, 2,…, k. The numbers x_i are called the
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A partition of a positive integer n is a representation of n as a sum of positive integers n = x₁ + x₂ +⋯+ x_k, x_i ≥ 1, i = 1, 2,…, k. The numbers x_i are called the
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partition
…and number theory, is the partition of a positive integer—that is, the number of ways that an integer n can be expressed as the sum of k smaller integers. For example, the number of ways of representing the number 7 as the sum of 3 smaller whole numbers (n =…
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…and number theory, is the partition of a positive integer—that is, the number of ways that an integer n can be expressed as the sum of k smaller integers. For example, the number of ways of representing the number 7 as the sum of 3 smaller whole numbers (n =…
Read More

Srinivasa Ramanujan
Further advances and philosophical reflections
…further advances, especially in the partition of numbers (the number of ways that a positive integer can be expressed as the sum of positive integers; e.g., 4 can be expressed as 4; 3 + 1; 2 + 2; 2 + 1 + 1; and 1 + 1 + 1 +…
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…further advances, especially in the partition of numbers (the number of ways that a positive integer can be expressed as the sum of positive integers; e.g., 4 can be expressed as 4; 3 + 1; 2 + 2; 2 + 1 + 1; and 1 + 1 + 1 +…
Read More