---
tag: M4H8KC15HB
kind: concept
name: Combinatorial number system
aliases: [combinadic, combinadics, gospers-hack, macaulay-representation-of-an-integer]
sources:
  - wikipedia: en:1113115@r1370497250
    locator: Combinatorial number system
    note: adopted
    mention: wikipedia://en/1113115
    sections: [Summary]
same_as:
  wikidata: Q5150829
  wikipedia: Combinatorial number system
inventory: [EXFTS7VC0J]
---

## Summary

In 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*<sub>*k*</sub> > ... > *c*<sub>2</sub> > *c*<sub>1</sub> ≥ 0 where each *c<sub>i</sub>* 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.