unordered partition

mathematics

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July 14, 2026

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In mathematics, a partition of a positive integer is a way to express that integer as a sum of smaller positive integers. These smaller 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 + 2.12

When considering unordered partitions, the order of the parts does not matter. These partitions can be standardized by listing the parts in descending order. The number of partitions of n into k parts is denoted by P_k(n). Tools like Ferrers' diagrams can be used to visualize and derive results about partitions, such as the relationship between partitions into k parts and partitions where k is the largest part.23

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combinatorics
Partitions
The theory of unordered partitions is much more difficult and has many interesting features. An unordered partition can be standardized by listing the parts in a decreasing order. Thus n = x₁ + x₂ +⋯+ x_k, x₁ ≥ x₂ ≥⋯≥ x_k ≥ 1. In what follows, partition will…
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The theory of unordered partitions is much more difficult and has many interesting features. An unordered partition can be standardized by listing the parts in a decreasing order. Thus n = x₁ + x₂ +⋯+ x_k, x₁ ≥ x₂ ≥⋯≥ x_k ≥ 1. In what follows, partition will…
Read More