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 ${\tbinom {n}{k}}$ 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

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.^[1] Indeed, a greedy algorithm finds the k-combination corresponding to N: take c_k maximal with ${\tbinom {c_{k}}{k}}\leq N$ , then take c_k−1 maximal with ${\tbinom {c_{k-1}}{k-1}}\leq N-{\tbinom {c_{k}}{k}}$ , 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.^[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 ${\tbinom {c_{i}}{i}}$ 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.

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 ${\tbinom {n}{k}}$ 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, ..., c_k} describes the number

2^{c_{1}}+2^{c_{2}}+\cdots +2^{c_{k}}

(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 = {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 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 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 k-combinations S strictly less than C. The number of possible choices is ${\tbinom {c_{i}}{i}}$ , which is therefore the number of combinations in group i; the total number of k-combinations strictly less than C then is

{\binom {c_{1}}{1}}+{\binom {c_{2}}{2}}+\cdots +{\binom {c_{k}}{k}},

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

Example

Suppose one wants to determine the 5-combination at position 72. The successive values of ${\tbinom {n}{5}}$ 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 ${\tbinom {n}{4}}$ 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 $72={\tbinom {8}{5}}+{\tbinom {6}{4}}+{\tbinom {3}{3}}$ , and the remaining values c_i will be the maximal ones with ${\tbinom {c_{i}}{i}}=0$ , namely c_i = i − 1. Thus we have found the 5-combination {8, 6, 3, 1, 0}.

National Lottery example

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

{\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}}.

References

^ Applied Combinatorial Mathematics, Ed. E. F. Beckenbach (1964), pp.27−30.
^ Generating Elementary Combinatorial Objects, Lucia Moura, U. Ottawa, Fall 2009
^ "Combinations — Sage 9.4 Reference Manual: Combinatorics".
^ Knuth, D. E. (2005), "Generating All Combinations and Partitions", The Art of Computer Programming, vol. 4, Fascicle 3, Addison-Wesley, pp. 5−6, ISBN 0-201-85394-9.
^ Pascal, Ernesto (1887), Giornale di Matematiche, vol. 25, pp. 45−49
^ McCaffrey, James (2004), Generating the mth Lexicographical Element of a Mathematical Combination, Microsoft Developer Network