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, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are represented as strictly decreasing sequences c_k > ... > c₂ > c₁ ≥ 0 where each c_i 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. The numbers less than ${\displaystyle {\tbinom {n}{k}}}$ 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

${\displaystyle N={\binom {c_{k}}{k}}+\cdots +{\binom {c_{2}}{2}}+{\binom {c_{1}}{1}}}$.

The fact that a unique sequence corresponds to any non-negative number N was first observed by D. H. Lehmer. Indeed, a greedy algorithm finds the k-combination corresponding to N: take c_k maximal with ${\displaystyle {\tbinom {c_{k}}{k}}\leq N}$, then take c_k−1 maximal with ${\displaystyle {\tbinom {c_{k-1}}{k-1}}\leq N-{\tbinom {c_{k}}{k}}}$, and so forth. Finding the number N, using the formula above, from the k-combination (c_k, ..., c₂, c₁) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; the operations are known by these names in most computer algebra systems, and in computational mathematics.

The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth, who also gives a much older reference; the term "combinadic" is introduced by James McCaffrey (without reference to previous terminology or work).

Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part ${\displaystyle {\tbinom {c_{i}}{i}}}$ 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 are software testing, sampling, quality control, and the analysis of lottery games.