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 ck > ... > c2 > c1 ≥ 0 where each ci 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]\displaystyle{ \tbinom nk }[/math] 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 (ck, ..., c2, c1) is given by

[math]\displaystyle{ 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.[1] Indeed, a greedy algorithm finds the k-combination corresponding to N: take ck maximal with [math]\displaystyle{ \tbinom{c_k}k\leq N }[/math], then take ck−1 maximal with [math]\displaystyle{ \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 (ck, ..., 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.[2][3]

The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth,[4] who also gives a much older reference;[5] the term "combinadic" is introduced by James McCaffrey[6] (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]\displaystyle{ \tbinom{c_i}i }[/math] of the number N represented by a "digit" ci 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.

## Contents

* 1 Ordering combinations
* 2 Place of a combination in the ordering
* 3 Finding the k-combination for a given number
* 3.1 Example
* 4 National Lottery example
* 5 See also
* 6 References
* 7 Further reading

## Ordering combinations

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]\displaystyle{ \tbinom nk }[/math] possible k-combinations of a set S of n elements. Choosing, for any n, {0, 1, ..., n − 1} 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 {0, 1, ..., n − 1} are less than those representing k-combinations not contained in {0, 1, ..., n − 1}, 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, ..., ck} describes the number

[math]\displaystyle{ 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) k-combinations.

## Place of a combination in the ordering

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 = {ck, ..., c2, c1} with ck > ... > 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 ci is absent from S, while ck, ..., ci+1 are present in S, and no other value larger than ci is. One can therefore group those k-combinations 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 ck, ..., ci+1 in S, and the remaining i elements of S must be chosen from the ci non-negative integers strictly less than ci; moreover any such choice will result in a k-combinations S strictly less than C. The number of possible choices is [math]\displaystyle{ \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]\displaystyle{ \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 number

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 ck 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 ck as the largest element is [math]\displaystyle{ \tbinom{c_k}k }[/math], and it has ci = i − 1 for all i < k (for this combination all terms in the expression except [math]\displaystyle{ \tbinom{c_k}k }[/math] are zero). Therefore ck is the largest number such that [math]\displaystyle{ \tbinom{c_k}k\leq N }[/math]. If k > 1 the remaining elements of the k-combination form the k − 1-combination corresponding to the number [math]\displaystyle{ N-\tbinom{c_k}k }[/math] in the combinatorial number system of degree k − 1, and can therefore be found by continuing in the same way for [math]\displaystyle{ N-\tbinom{c_k}k }[/math] and k − 1 instead of N and k.

### Example

Suppose one wants to determine the 5-combination at position 72. The successive values of [math]\displaystyle{ \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 4-combination at position 72 − 56 = 16. The successive values of [math]\displaystyle{ \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 16 − 15 = 1 one finds c3 = 3, which uses up the final unit; this establishes [math]\displaystyle{ 72=\tbinom85+\tbinom64+\tbinom33 }[/math], and the remaining values ci will be the maximal ones with [math]\displaystyle{ \tbinom{c_i}i=0 }[/math], namely ci = i − 1. Thus we have found the 5-combination {8, 6, 3, 1, 0}.

## National Lottery example

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

[math]\displaystyle{ \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]

## See also

* Factorial number system (also called factoradics)
* Primorial number system
* Asymmetric numeral systems \- also e.g. of combination to natural number, widely used in data compression

## References

1. ↑ Applied Combinatorial Mathematics, Ed. E. F. Beckenbach (1964), pp.27−30.
2. ↑ http://www.site.uottawa.ca/~lucia/courses/5165-09/GenCombObj.pdf[bare URL PDF]
3. ↑ "Combinations — Sage 9.4 Reference Manual: Combinatorics". https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/combination.html.
4. ↑ "Generating All Combinations and Partitions", The Art of Computer Programming, 4, Fascicle 3, Addison-Wesley, 2005, pp. 5−6, ISBN 0-201-85394-9 .
5. ↑ Pascal, Ernesto (1887), Giornale di Matematiche, 25, pp. 45−49
6. ↑ Generating the mth Lexicographical Element of a Mathematical Combination, Microsoft Developer Network, 2004, https://docs.microsoft.com/en-us/previous-versions/visualstudio/aa289166(v=vs.70)

## Further reading

* Huneke, Craig; Swanson, Irena (2006), "Appendix 5", Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, http://people.reed.edu/~iswanson/book/index.html
* Caviglia, Giulio (2005), "A theorem of Eakin and Sathaye and Green's hyperplane restriction theorem", Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects, CRC Press, ISBN 978-1-420-02832-4, https://books.google.com/books?id=Z0RfAU2Vw6YC&pg=PA2
* Green, Mark (1989), "Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann", Algebraic Curves and Projective Geometry, Lecture Notes in Mathematics, 1389, Springer, pp. 76–86, doi:10.1007/BFb0085925, ISBN 978-3-540-48188-1

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