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 <math>\tbinom nk</math> correspond to all Template:Nowrap of Template:Nowrap}. 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

<math>N=\binom{c_k}k+\cdots+\binom{c_2}2+\binom{c_1}1</math>.

The fact that a unique sequence corresponds to any non-negative number N was first observed by D. H. Lehmer.<ref>Applied Combinatorial Mathematics, Ed. E. F. Beckenbach (1964), pp.27−30.</ref> Indeed, a greedy algorithm finds the k-combination corresponding to N: take c_k maximal with <math>\tbinom{c_k}k\leq N</math>, then take c_k−1 maximal with <math>\tbinom{c_{k-1}}{k-1}\leq N - \tbinom{c_k}k</math>, 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.<ref>Generating Elementary Combinatorial Objects, Lucia Moura, U. Ottawa, Fall 2009</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth,<ref>Template:Citation.</ref> who also gives a much older reference;<ref>Template:Citation</ref> the term "combinadic" is introduced by James McCaffrey<ref>Template:Citation</ref> (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 <math>\tbinom{c_i}i</math> 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 Template:Nowrap 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.

Ordering combinationsEdit

A k-combination of a set S is a subset of S with k (distinct) elements. The main purpose of the combinatorial number system is to provide a representation, each by a single number, of all <math>\tbinom nk</math> possible k-combinations of a set S of n elements. Choosing, for any n, Template:Nowrap} as such a set, it can be arranged that the representation of a given k-combination C is independent of the value of n (although n must of course be sufficiently large); in other words considering C as a subset of a larger set by increasing n will not change the number that represents C. Thus for the combinatorial number system one just considers C as a k-combination of the set N of all natural numbers, without explicitly mentioning n.

In order to ensure that the numbers representing the k-combinations of Template:Nowrap} are less than those representing k-combinations not contained in Template:Nowrap}, the k-combinations must be ordered in such a way that their largest elements are compared first. The most natural ordering that has this property is lexicographic ordering of the decreasing sequence of their elements. So comparing the 5-combinations C = {0,3,4,6,9} and C′ = {0,1,3,7,9}, one has that C comes before C′, since they have the same largest part 9, but the next largest part 6 of C is less than the next largest part 7 of C′; the sequences compared lexicographically are (9,6,4,3,0) and (9,7,3,1,0).

Another way to describe this ordering is view combinations as describing the k raised bits in the binary representation of a number, so that C = {c₁, ..., c_k} describes the number

<math>2^{c_1}+2^{c_2}+\cdots+2^{c_k}</math>

(this associates distinct numbers to all finite sets of natural numbers); then comparison of k-combinations can be done by comparing the associated binary numbers. In the example C and C′ correspond to numbers 1001011001₂ = 601₁₀ and 1010001011₂ = 651₁₀, which again shows that C comes before C′. This number is not however the one one wants to represent the k-combination with, since many binary numbers have a number of raised bits different from k; one wants to find the relative position of C in the ordered list of (only) Template:Nowrap.

Place of a combination in the orderingEdit

The number associated in the combinatorial number system of degree k to a k-combination C is the number of k-combinations strictly less than C in the given ordering. This number can be computed from C = {c_k, ..., c₂, c₁} with c_k > ... > c₂ > c₁ as follows.

From the definition of the ordering it follows that for each k-combination S strictly less than C, there is a unique index i such that c_i is absent from S, while c_k, ..., c_i+1 are present in S, and no other value larger than c_i is. One can therefore group those Template:Nowrap S according to the possible values 1, 2, ..., k of i, and count each group separately. For a given value of i one must include c_k, ..., c_i+1 in S, and the remaining i elements of S must be chosen from the c_i non-negative integers strictly less than c_i; moreover any such choice will result in a Template:Nowrap S strictly less than C. The number of possible choices is <math>\tbinom{c_i}i</math>, which is therefore the number of combinations in group i; the total number of k-combinations strictly less than C then is

<math>\binom{c_1}1+\binom{c_2}2+\cdots+\binom{c_k}k,</math>

and this is the index (starting from 0) of C in the ordered list of k-combinations.

Obviously there is for every N ∈ N exactly one k-combination at index N in the list (supposing k ≥ 1, since the list is then infinite), so the above argument proves that every N can be written in exactly one way as a sum of k binomial coefficients of the given form.

Finding the k-combination for a given numberEdit

The given formula allows finding the place in the lexicographic ordering of a given k-combination immediately. The reverse process of finding the k-combination at a given place N requires somewhat more work, but is straightforward nonetheless. By the definition of the lexicographic ordering, two k-combinations that differ in their largest element c_k will be ordered according to the comparison of those largest elements, from which it follows that all combinations with a fixed value of their largest element are contiguous in the list. Moreover the smallest combination with c_k as the largest element is <math>\tbinom{c_k}k</math>, and it has c_i = i − 1 for all i < k (for this combination all terms in the expression except <math>\tbinom{c_k}k</math> are zero). Therefore c_k is the largest number such that <math>\tbinom{c_k}k\leq N</math>. If k > 1 the remaining elements of the k-combination form the Template:Nowrap-combination corresponding to the number <math>N-\tbinom{c_k}k</math> in the combinatorial number system of degree Template:Nowrap, and can therefore be found by continuing in the same way for <math>N-\tbinom{c_k}k</math> and Template:Nowrap instead of N and k.

ExampleEdit

Suppose one wants to determine the 5-combination at position 72. The successive values of <math>\tbinom n5</math> for n = 4, 5, 6, ... are 0, 1, 6, 21, 56, 126, 252, ..., of which the largest one not exceeding 72 is 56, for n = 8. Therefore c₅ = 8, and the remaining elements form the Template:Nowrap at position Template:Nowrap. The successive values of <math>\tbinom n4</math> for n = 3, 4, 5, ... are 0, 1, 5, 15, 35, ..., of which the largest one not exceeding 16 is 15, for n = 6, so c₄ = 6. Continuing similarly to search for a 3-combination at position Template:Nowrap one finds c₃ = 3, which uses up the final unit; this establishes <math>72=\tbinom85+\tbinom64+\tbinom33</math>, and the remaining values c_i will be the maximal ones with <math>\tbinom{c_i}i=0</math>, namely Template:Nowrap. Thus we have found the 5-combination Template:Nowrap}.

National Lottery exampleEdit

For each of the <math>\binom{49}6</math> lottery combinations c₁ < c₂ < c₃ < c₄ < c₅ < c₆ , there is a list number N between 0 and <math>\binom{49}6 - 1</math> which can be found by adding

<math> \binom{49-c_1} 6 + \binom{49-c_2} 5 + \binom{49-c_3} 4 + \binom{49-c_4} 3 + \binom{49-c_5} 2 + \binom{49-c_6} 1. </math>

ReferencesEdit