{
  "aliases": [
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  "body": "\n## Summary\n\nIn mathematics, and in particular in [[combinatorics|0JKW7W7XFR]], the **combinatorial number system** of degree *k* (for some positive [[integer|0C4JS9RF5M]] *k*), also referred to as **combinadics**, or the **Macaulay representation of an integer**, is a correspondence between [[natural numbers|0MWKJ44V5A]] (taken to include 0) *N* and *k*-[[combinations|098PHYBTV9]]. The combinations are represented as [[strictly decreasing|0G9EKM0E15]] sequences *c*\u003csub\u003e*k*\u003c/sub\u003e \u003e ... \u003e *c*\u003csub\u003e2\u003c/sub\u003e \u003e *c*\u003csub\u003e1\u003c/sub\u003e ≥ 0 where each *c\u003csub\u003ei\u003c/sub\u003e* corresponds to the index of a chosen element in a given *k*-combination. Distinct numbers correspond to distinct *k*-combinations, and produce them in [[lexicographic order|0B68KNPGCG]]. The numbers less than $\\tbinom nk$ correspond to all *k*-combinations of {0, 1, ..., *n* − 1}. The correspondence does not depend on the size *n* of the set that the *k*-combinations are taken from, so it can be interpreted as a map from **N** to the *k*-combinations taken from **N**; in this view the correspondence is a bijection.\n",
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