Circular Permutations

A circular permutation is an arrangement of $ n $ elements around a circle, in which there is no distinguished starting position. Two arrangements are considered identical if one can be obtained from the other by rotating the circle.

Mathematically, circular permutations correspond to equivalence classes of linear permutations under the action of the cyclic group $ C_n $

A classical (linear) permutation has a starting position and an ending position. In a circular permutation, no starting position is distinguished.

To visualise this difference, imagine a circle: cutting the circle at different points produces several different linear lists, but only one circular arrangement.

For $ n $ distinct elements, the number of circular permutations is: $ (n-1)! $

Indeed, by fixing one element arbitrarily, the remaining $ n-1 $ elements can be permuted freely. There are therefore $ (n-1)! $ arrangements.

For distinct elements, it is possible to choose a starting point for each rotation class.

1 - fix one element

2 - generate all permutations of the remaining $ n-1 $ elements using backtracking

3 - add the fixed element at the beginning of each generated permutation

Each resulting permutation then represents a different circular permutation, without duplicates.

Circular permutations arise in many problems where no starting position is distinguished.

— seating guests around a round table

— arranging cyclic sequences or patterns

— counting necklaces of beads or coloured objects

— studying circular biological sequences (some bacterial genomes)

— combinatorial problems involving cycles or periodic configurations