{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "001CX45SWQ"
    },
    {
      "field": "uses",
      "target": "0K78D81EF7"
    },
    {
      "field": "uses",
      "target": "0XFZ1BMF5Q"
    }
  ],
  "body": "\n## Definition\n\nLet $n$ be a natural number.\nA partition of $n$ is a finite multiset of positive integers whose sum is $n$.\nIts elements are the parts of the partition.\n\nEquivalently, a partition of $n$ is a finite sequence $\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_k$ of positive integers with $\\lambda_1 + \\dots + \\lambda_k = n$.\n\n## Remarks\n\nThe order of the parts does not matter. A sum in which the order matters is a [[composition|0J8ANSYKAC]].\nThe $5$ partitions of $4$ are $4$, $3 + 1$, $2 + 2$, $2 + 1 + 1$ and $1 + 1 + 1 + 1$.\nThe number $0$ has exactly one partition, the empty partition.\nThe number of partitions of $n$ is the [[partition function|001CX45SWQ]] $p(n)$, and a [[Young diagram|0K78D81EF7]] draws a partition.\n\n## Explanations\n\n### L1 human\n\nA partition of $n$ is a way to write $n$ as a sum of positive integers when the order of the summands does not matter.\nSo $3 + 1$ and $1 + 3$ are the same partition of $4$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0JV30XN73Z",
    "kind": "definition",
    "msc": [
      "05A17"
    ],
    "name": "Integer partition",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Integer_partition"
      }
    ],
    "tag": "0404R9P9J3"
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0404R9P9J3/",
  "links": [
    {
      "field": "defines",
      "target": "0JV30XN73Z"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0404R9P9J3.md",
  "msc": [
    "05A17"
  ],
  "name": "Integer partition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/040/0404R9P9J3.md",
  "status": null,
  "tag": "0404R9P9J3",
  "uri": "m2://0404R9P9J3",
  "url": "https://math2.tamnd.com/wiki/integer-partition-definition/",
  "writing": {
    "tag": "0404R9P9J3",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}