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Circular Permutation


The number of ways to arrange n distinct objects along a fixed (i.e., cannot be picked up out of the plane and turned over) circle is

 P_n=(n-1)!.

The number is (n-1)! instead of the usual factorial n! since all cyclic permutations of objects are equivalent because the circle can be rotated.

CircularPermutations

For example, of the 3!=6 permutations of three objects, the (3-1)!=2 distinct circular permutations are {1,2,3} and {1,3,2}. Similarly, of the 4!=24 permutations of four objects, the (4-1)!=6 distinct circular permutations are {1,2,3,4}, {1,2,4,3}, {1,3,2,4}, {1,3,4,2}, {1,4,2,3}, and {1,4,3,2}. Of these, there are only three free permutations (i.e., inequivalent when flipping the circle is allowed): {1,2,3,4}, {1,2,4,3}, and {1,3,2,4}. The number of free circular permutations of order n is P_n^'=1 for n=1, 2, and

 P_n^'=1/2(n-1)!

for n>=3, giving the sequence 1, 1, 1, 3, 12, 60, 360, 2520, ... (OEIS A001710).


See also

Cyclic Permutation, Factorial, Permutation, Prime Circle

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References

Sloane, N. J. A. Sequence A001710/M2933 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Circular Permutation

Cite this as:

Weisstein, Eric W. "Circular Permutation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CircularPermutation.html

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