S(r, n), represents the number of ways that we can arrange r objects around indistinguishable circles of length n, and every circle n must have at least one object around it.
Input: r = 9, n = 2 Output: 109584 Input: r = 6, n = 3 Output: 225
The special cases are:
- S(r, 0) = 0, trivial.
- S(r, 1) represents the circular permutation which is equal to (r – 1)!
- S(r, n) where r = n, equals 1.
- S(r, r -1) = rC2
An important identity of the Stirling numbers that S(r, n) = S(r – 1, n – 1) + (r – 1) * S(r – 1, n)
Approach: For simplicity, denote the r distinct objects by 1, 2, …, r. Consider the object “1”. In any arrangement of the objects, either
- “1” is the only object in a circle or
- “1” is mixed with others in a circle.
In case 1, there are s(r – 1, n – 1) ways to form such arrangements. In case 2, first of all, the r — 1 objects 2, 3, …, r are put in n circles in s(r — 1, n) ways; then “1” can be placed in one of the r — 1 distinct spaces to the “immediate right” of the corresponding r — 1 distinct objects. By multiplication principle, there are (r — 1)s(r — 1, n) ways to form such arrangements in case 2. The identity now follows from the definition of s(r, n) and addition principle.
Using the initial values S(0, 0) = 1, s(r, 0) = 0 for r > 1 and s(r, 1) = (r — 1)! for r > 1, and applying the identity we proved, we can easily get the Stirling number by computing it in a recursive way.
In the code we have three functions that are used to generate the Stirling numbers, which are nCr(n, r), which is a function to compute what we call (n – choose – r), the number of ways we can take r objects from n objects without the importance of orderings. factorial (int n) is, unsurprisingly, used to compute the factorial of a number n. The function Stirling number(r, n) works recursively using the four base cases discussed above and then recursing using the identity we proved.
Below is the implementation of the above approach:
The Stirling Number s(9, 2) is : 109584
Please refer Stirling numbers of the first kind to read more about the Stirling numbers.
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