Combinatorial Number System

In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, is a correspondence between natural numbers (taken to include 0) N and k-combinations, represented as strictly decreasing sequences c_k > ... > c₂ > c₁ ≥ 0. Since the latter are strings of numbers, one can view this as a kind of numeral system for representing N, although the main utility is representing a k-combination by N rather than the other way around. Distinct numbers correspond to distinct k-combinations, and produce them in lexicographic order; moreover the numbers less than 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.

The number N corresponding to (c_k,...,c₂,c₁) is given by

The fact that a unique sequence so corresponds to any number N was observed by D.H. Lehmer. Indeed a greedy algorithm finds the k-combination corresponding to N: take c_k maximal with, then take c_{k − 1} maximal with, and so forth. The originally used term "combinatorial representation of integers" is shortened to "combinatorial number system" by Knuth, who also gives a much older reference; the term "combinadic" is introduced by James McCaffrey with a computer program implementation.

Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part of the number N represented by a "digit" c_i is not obtained from it by simply multiplying by a place value.

The main application of the combinatorial number system is that it allows rapid computation of the k-combination that is at a given position in the lexicographic ordering, without having to explicitly list the k-combinations preceding it; this allows for instance random generation of k-combinations of a given set. Enumeration of k-combinations has many applications, among which software testing, sampling, quality control, and the analysis of lottery games.

This is also known as "rank" ("ranking" and "unranking"), and is known by that name in most CAS software and in computational mathematics.