Understanding Circular Permutation: Step-by-Step Guide

Circular permutation concerns the enumeration of ordered arrangements of distinct objects placed around a circle, where linear shifts correspond to the same configuration due to the circle's symmetry.

Enumeration of Distinct Circular Arrangements of $n$ Objects

Let $n$ be a positive integer with $n \geq 2$. Consider $n$ distinct objects that are to be arranged on the circumference of a fixed circle. In a linear arrangement, the number of possible orderings is $n!$, as each object can independently occupy any of the $n$ positions.

When arranged on a circle, the absence of a natural starting point leads to equivalence between orderings that differ only by rotation. Thus, all permutations obtained by rotating a fixed linear arrangement are considered identical as circular arrangements.

For any linear permutation, there exist $n$ rotational variants—one for each object occupying each fixed position on the circle. All $n$ circular placements, derived by cyclic rotation of a fixed linear arrangement, produce the same circular permutation.

The total number of distinct circular permutations of $n$ distinct objects in a fixed circle is therefore given by:

$\displaystyle \text{Number of circular permutations} = \dfrac{n!}{n} = (n-1)!$

Result: The number of distinct arrangements of $n$ distinct objects in a circle, when rotations are considered identical and reflections are distinct, is $(n-1)!$.

Canonical Labeling and Justification of the $(n-1)!$ Formula

To formalize the above result, fix one object at a chosen reference position on the circle. The arrangement problem thus reduces to permuting the remaining $n-1$ objects in a linear order around the remaining positions, taken in a fixed direction (e.g., clockwise).

The number of distinct ways to arrange the $n-1$ objects in the remaining positions is $(n-1)!$ by definition of the factorial function, since each object must occupy a unique position, and all linear orders are allowed.

Stepwise expansion: Suppose the objects are labeled $A_1$, $A_2$, $\ldots$, $A_n$. Fix $A_1$ at an arbitrary position on the circle. The first available position (adjacent to $A_1$) can be filled by any of the remaining $n-1$ objects; the next position can be filled by any of the remaining $n-2$ objects, and so on until all $n-1$ positions are filled. Therefore,

$\displaystyle \text{Total circular arrangements} = (n-1) \times (n-2) \times \cdots \times 2 \times 1 = (n-1)!$

Reflections and the Case of Indistinguishable Arrangements under Rotation and Reflection

If the circle is not fixed in the plane—that is, arrangements indistinguishable by either rotation or reflection (as in the case of a necklace)—then each arrangement is equivalent to its mirror image. In such cases, the total number of distinct circular permutations is reduced by a further factor of 2, yielding:

$\displaystyle \text{Number of circular arrangements (rotation and reflection indistinguishable)} = \dfrac{(n-1)!}{2}$

Special Case: This case applies to objects arranged on a freely invertible structure, such as a necklace, where clockwise and anticlockwise arrangements represent the same physical configuration.

Worked Example: Circular Arrangement of Distinct Objects

Given: Four distinct books are to be arranged around a round table.

Substitution: Here $n = 4$ and only arrangements differing by rotation are considered identical. No reflection equivalence is imposed (table is fixed).

Simplification: The number of distinct circular permutations is $(n-1)! = (4-1)! = 3! = 6$.

Final result: There are $6$ distinct ways to place the books around the table.

For foundational concepts on permutations, see Permutations And Combinations.

Circular Permutations with Certain Elements Fixed

If one or more elements are constrained to occupy specific positions relative to each other, treat the group as a single composite unit. For example, if $k$ specific objects must always be together, treat these as one block. The circular arrangement then involves $(n - k + 1)$ entities: the block plus the remaining $n-k$ objects.

The number of ways to arrange the group around a circle is $(n-k)!$ when considering only rotations, while the internal order of the $k$ grouped objects can be arranged in $k!$ ways. Thus, the total number of circular permutations becomes:

$\displaystyle \text{Total circular arrangements with $k$ objects together} = (n-k)! \times k!$

Example: Seven people, with three friends who must sit together, can be arranged around a circular table in $(7 - 3)! \times 3! = 4! \times 6 = 24 \times 6 = 144$ ways.

Circular Arrangements with Restrictions on the Number Taken at a Time

If $n$ objects are to be arranged around a circle taking $r$ at a time ($2 \leq r \leq n$), the number of linear permutations is $P(n, r) = n! / (n - r)!$, where $P(n, r)$ denotes the number of linear arrangements of $r$ objects selected from $n$.

For circular arrangements, as $r$ positions can be rotated to coincide, each configuration appears $r$ times among the linear arrangements. Therefore,

$\displaystyle \text{Circular permutations of $n$ objects taken $r$ at a time} = \dfrac{P(n, r)}{r} = \dfrac{n!}{(n - r)! \cdot r}$

Example: Twelve people are to be seated on five chairs arranged in a circle. The number of possible arrangements is:

$\displaystyle \dfrac{12!}{(12 - 5)! \cdot 5} = \dfrac{12 \times 11 \times 10 \times 9 \times 8}{5} = \dfrac{95040}{5} = 19008$

Final result: There are $19,\!008$ distinct circular seating arrangements of twelve people on five chairs.

For additional arrangement scenarios, refer to Arrangement Of Different Objects.

Relation between Linear and Circular Permutations

A linear permutation of $n$ objects accounts for all possible orderings along a straight line and equals $n!$. Each circular permutation can be mapped to $n$ equivalent linear arrangements by choosing each object as the starting point. Thus,

$n! = n \cdot \text{(number of distinct circular permutations)}$

Therefore, the circular permutation count is:

$\dfrac{n!}{n} = (n-1)!$

For theory on permutation computation, visit Permutation And Combination.

Frequently Needed Formulas and Results in Circular Permutation

Formula 1: Number of ways to arrange $n$ distinct objects in a fixed circle, wherein only rotations are considered identical:

$(n-1)!$

Formula 2: Number of ways to arrange $n$ distinct objects in a circle where both rotations and reflections are considered identical:

$\dfrac{(n-1)!}{2}$

Formula 3: Number of ways to arrange $n$ objects around a circle taking $r$ at a time:

$\dfrac{n!}{(n-r)! \cdot r}$

Formula 4: Number of ways to arrange $n$ objects around a circle with $k$ together:

$(n-k)! \times k!$

For challenging permutation exercises, see Permutation And Combination Problems.

Special Cases: All Objects Identical or Some Objects Identical

If all $n$ objects are identical, then all circular arrangements are indistinguishable. Thus, the number of distinct arrangements is $1$.

If $n$ objects consist of $n_1$ identical of one kind, $n_2$ identical of another, and so on up to $n_k$ kinds, the number of linear arrangements is:

$\dfrac{n!}{n_1! n_2! \cdots n_k!}$

The corresponding number of circular permutations is:

$\dfrac{1}{n} \cdot \dfrac{n!}{n_1! n_2! \cdots n_k!}$

For extensive practice, consult Permutations And Combinations Practice Paper.