Necklace Permutation Calculator

Count necklace permutations (n−1)!/2 for n items arranged in a circle where flips are treated as identical, compared with circular and linear permutations.

Input

Calculate necklace permutations (n−1)! divided by 2, arranging n distinct items in a circle where flips (reflections) are treated as identical.

Number of items n

Enter an integer of 3 or more.

Result

Necklace permutations for n equal to 5

Circular (n−1)!

Linear n!

120

Ratio to circular

one half

The necklace permutation is the circular permutation (n−1)! halved by reflection, giving (n−1)! divided by 2. It counts arrangements that are the same under rotation and reflection.

How it works

A necklace permutation counts arrangements of n distinct items in a circle where two arrangements are the same if one is a rotation or a flip (reflection) of the other. It equals (n−1)! divided by 2, and is meaningful for n of 3 or more.
A circular permutation, where only rotations are treated as identical, gives (n−1)! arrangements. The necklace permutation halves this by also treating flips as identical, giving (n−1)! divided by 2.
For example, making a necklace from 5 distinct beads gives (5−1)! = 24 circular arrangements; removing mirror duplicates leaves 24 divided by 2 = 12 necklace permutations.
A plain linear permutation has n! arrangements. Closing it into a circle removes the rotations, and accounting for flips halves it again.
Exact integer arithmetic supports large values; when the result has many digits, the number of decimal digits is shown instead.

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