Combinatorial number system

This online calculator represents given natural number as sequences of k-combinations, referred as the combinatorial number system of degree k (for some positive integer k), aka combinadics

This online calculator converts natural number from decimal number system to so-called combinadics, or combinatorial number system of degree k. In this system, the number N is represented as sum of binomials:

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

and sequence of strictly decreasing coefficients c_k > \dots > c_2 > c_1 \geqslant 0 is the combinatorial representaton of N in combinatorial number system of degree k.

This representation is actually a bijection between natural number N and k-combination taken from N, so you can generate the k-combination at index N, which has some useful applications.

How to convert

You should enter a number to convert and desired combinatorial number system degree in the fields below, and the calculator shows you the representation and detail of conversion.

The conversion algorithm is described below the calculator.

PLANETCALC, Combinatorial number system

Combinatorial number system

Combinadics representation

Algorithm to find the k-combination for a given number

Finding the k-combination for a given number is called unranking and there is a simple greedy algorithm to do it:

  1. Find maximum c_k which holds {\tbinom {c_{k}}{k}}\leq N
  2. Find maximum c_{k-1} which holds {\tbinom {c_{k-1}}{k-1}}\leq N - {\tbinom {c_{k}}{k}}
  3. ...
  4. Find maximum c_1 which holds {\tbinom {c_1}{1}}\leq N - {\tbinom {c_{k}}{k}} - {\tbinom {c_{k-1}}{k-1}} - \cdots - {\tbinom {c_2}{2}}

Note that {\tbinom {c_{i}}{i}} = 0 if c_i < i. It is used to pad sum with zeroes, as you can see in the example above. In such cases, algorithm assigns c_i the maximum possible value: i-1

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PLANETCALC, Combinatorial number system