Conjugacy class size formula in symmetric group

Statement

Suppose $n$ is a natural number and $\lambda$ is an unordered integer partition of $n$ such that $\lambda$ has $a_j$ parts of size $j$ for each $j$ . In other words, there are $a_1$ $1$ s, $a_2$ $2$ s, $a_3$ $3$ s, and so on. Let $c$ be the conjugacy class in the symmetric group of degree $n$ comprising the elements whose cycle type is $\lambda$ , i.e., those elements whose cycle decomposition has $a_j$ cycles of length $j$ for each $j$ . Then:

$\! |c| = \frac{n!}{\prod_j (j)^{a_j}(a_j!)}$

Note that those $j$ where $a_j = 0$ contribute a $1$ in the denominator and can be ignored from the product, while for those $j$ where $a_j = 1$ , the $a_j!$ term can be omitted.

Equivalently, if $C$ is the centralizer of any element of $c$ , then:

$\! |C| = \prod_j (j)^{a_j}(a_j!)$

These are equivalent because size of conjugacy class equals index of centralizer, which follows from the identification of the conjugacy class with the left coset space of the centralizer via the action of the group on itself as automorphisms by conjugation.

Related facts

Examples

Illustrative examples

For instance, consider $n = 23$ with the partition $23 = 3 + 3 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1$ . There are four 3s, three 2s, and five 1s. An example element with this cycle type is given by the cycle decomposition:

$\! (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14)(15,16)(17,18)$

The size of the conjugacy class corresponding to this partition is:

$\! \frac{23!}{[(3)^4(4!)][(2)^3(3!)][(1)^5(5!)]}$

Here's another example: $13 = 5 + 4 + 4$ . There is one 5 and two 4s, and we get:

$\! \frac{13!}{[(5)^1(1!)][(4)^2(2!)]}$

When a particular $j$ has $a_j = 1$ (i.e., it occurs only once in the partition) then the corresponding term divided is $j^1(1!) = j$ , so the above can be written more briefly:

$\! \frac{13!}{[5][(4)^2(2!)]}$

Similarly, consider $15 = 5 + 4 + 3 + 3$ . We get:

$\! \frac{15!}{[5][4][(3)^2(2!)]}$

Comprehensive treatment of small degrees

In the right column links in the table below, you can see tabulated information on the sizes of conjugacy classes, as well as how the formula is applied to the cycle sizes to compute each specific size. The cases $n = 3,4,5$ are embedded below.

Degree	Symmetric group	List of conjugacy class sizes	Element structure page	Section on conjugacy class structure interpreted as symmetric group
3	symmetric group:S3	1,2,3	element structure of symmetric group:S3	element structure of symmetric group:S3#Interpretation as symmetric group
4	symmetric group:S4	1,3,6,6,8	element structure of symmetric group:S4	element structure of symmetric group:S4#Interpretation as symmetric group
5	symmetric group:S5	1,10,15,20,20,24,30	element structure of symmetric group:S5	element structure of symmetric group:S5#Interpretation as symmetric group
6	symmetric group:S6	1,15,15,40,40,45,90,90,120,120,144	element structure of symmetric group:S6	element structure of symmetric group:S6#Interpretation as symmetric group
7	symmetric group:S7	1,21,70,105,105,210,210,280, 420,420,504,504,630,720,840	element structure of symmetric group:S7	element structure of symmetric group:S7#Interpretation as symmetric group
8	symmetric group:S8	1, 28, 105, 112, 210, 420, 420, 1120, 1120, 1120, 1260, 1260, 1344, 1680, 2520, 2688, 3360, 3360, 3360, 4032, 5040, 5760	element structure of symmetric group:S8	element structure of symmetric group:S8#Interpretation as symmetric group

$n = 3$

Partition	Partition in grouped form	Verbal description of cycle type	Elements with the cycle type in cycle decomposition notation	Elements with the cycle type in one-line notation	Size of conjugacy class	Formula for size	Even or odd? If even, splits? If splits, real in alternating group?	Element order	Formula calculating element order
1 + 1 + 1	1 (3 times)	three fixed points	$()$ -- the identity element	123	1	$\! \frac{3!}{(1)^3(3!)}$	even; no	1	$\operatorname{lcm} \{ 1, 1, 1 \}$
2 + 1	2 (1 time), 1 (1 time)	transposition in symmetric group:S3: one 2-cycle, one fixed point	$(1,2)$ , $(1,3)$ , $(2,3)$	213, 321, 132	3	$\! \frac{3!}{(2)(1)}$	odd	2	$\operatorname{lcm} \{ 2,1 \}$
3	3 (1 time)	3-cycle in symmetric group:S3: one 3-cycle	$(1,2,3)$ , $(1,3,2)$	231, 312	2	$\! \frac{3!}{3}$	even; yes; no	3	$\operatorname{lcm} \{ 3 \}$
Total (3 rows -- 3 being the number of unordered integer partitions of 3)	--	--	--	--	6 (equals 3!, the size of the symmetric group)	--	odd: 3 even;no: 1 even; yes; no: 2	order 1: 1, order 2: 3, order 3: 2	--

$n = 4$

Partition	Partition in grouped form	Verbal description of cycle type	Elements with the cycle type	Size of conjugacy class	Formula for size	Even or odd? If even, splits? If splits, real in alternating group?	Element order	Formula calculating element order
1 + 1 + 1 + 1	1 (4 times)	four cycles of size one each, i.e., four fixed points	$()$ -- the identity element	1	$\! \frac{4!}{(1)^4(4!)}$	even; no	1	$\operatorname{lcm}\{ 1,1,1,1 \}$
2 + 1 + 1	2 (1 time), 1 (2 times)	one transposition (cycle of size two), two fixed points	$(1,2)$ , $(1,3)$ , $(1,4)$ , $(2,3)$ , $(2,4)$ , $(3,4)$	6	$\! \frac{4!}{[(2)^1(1!)][(1)^2(2!)]}$ , also $\binom{4}{2}$	odd	2	$\operatorname{lcm}\{2,1,1 \}$
2 + 2	2 (2 times)	double transposition: two cycles of size two	$(1,2)(3,4)$ , $(1,3)(2,4)$ , $(1,4)(2,3)$	3	$\! \frac{4!}{(2)^2(2!)}$	even; no	2	$\operatorname{lcm}\{2,2 \}$
3 + 1	3 (1 time), 1 (1 time)	one 3-cycle, one fixed point	$(1,2,3)$ , $(1,3,2)$ , $(2,3,4)$ , $(2,4,3)$ , $(3,4,1)$ , $(3,1,4)$ , $(4,1,2)$ , $(4,2,1)$	8	$\! \frac{4!}{[(3)^1(1!)][(1)^1(1!)]}$ or $\! \frac{4!}{(3)(1)}$	even; yes; no	3	$\operatorname{lcm}\{3,1 \}$
4	4 (1 time)	one 4-cycle, no fixed points	$(1,2,3,4)$ , $(1,2,4,3)$ , $(1,3,2,4)$ , $(1,3,4,2)$ , $(1,4,2,3)$ , $(1,4,3,2)$	6	$\! \frac{4!}{(4)^1(1!)}$ or $\frac{4!}{4}$	odd	4	$\operatorname{lcm} \{ 4 \}$
Total (5 rows, 5 being the number of unordered integer partitions of 4)	--	--	--	24 (equals 4!, the order of the whole group)	--	odd: 12 (2 classes) even; no: 4 (2 classes) even; yes; no: 8 (1 class)	order 1: 1 (1 class) order 2: 9 (2 classes) order 3: 8 (1 class) order 4: 6 (1 class)	--

$n = 5$

Partition	Partition in grouped form	Verbal description of cycle type	Representative element with the cycle type	Size of conjugacy class	Formula calculating size	Even or odd? If even, splits? If splits, real in alternating group?	Element order	Formula calcuating element order
1 + 1 + 1 + 1 + 1	1 (5 times)	five fixed points	$()$ -- the identity element	1	$\! \frac{5!}{(1)^5(5!)}$	even; no	1	$\operatorname{lcm}\{1 \}$
2 + 1 + 1 + 1	2 (1 time), 1 (3 times)	transposition: one 2-cycle, three fixed point	$(1,2)$	10	$\! \frac{5!}{[(2)^1(1!)][(1)^3(3!)]}$ or $\! \frac{5!}{(2)(1)^3(3!)}$ , also $\binom{5}{2}$ in this case	odd	2	$\operatorname{lcm}\{2,1 \}$
3 + 1 + 1	3 (1 time), 1 (2 times)	one 3-cycle, two fixed points	$(1,2,3)$	20	$\! \frac{5!}{[(3)^1(1!)][(1)^2(2!)]}$ or $\! \frac{5!}{(3)(1)^2(2!)}$	even; no	3	$\operatorname{lcm}\{3,1\}$
2 + 2 + 1	2 (2 times), 1 (1 time)	double transposition: two 2-cycles, one fixed point	$(1,2)(3,4)$	15	$\frac{5!}{[2^2(2!)][1^1(1!)]}$ or $\! \frac{5!}{2^2(2!)(1)}$	even; no	2	$\operatorname{lcm}\{2,1 \}$
4 + 1	4 (1 time), 1 (1 time)	one 4-cycle, one fixed point	$(1,2,3,4)$	30	$\! \frac{5!}{[4^1(1!)][1^1(1!)]}$ or $\! \frac{5!}{(4)(1)}$	odd	4	$\operatorname{lcm}\{4,1\}$
3 + 2	3 (1 time), 2 (1 time)	one 3-cycle, one 2-cycle	$(1,2,3)(4,5)$	20	$\! \frac{5!}{[3^1(1!)][2^1(1!)]}$ or $\! \frac{5!}{(3)(2)}$	odd	6	$\operatorname{lcm}\{3,2 \}$
5	5 (1 time)	one 5-cycle	$(1,2,3,4,5)$	24	$\frac{5!}{5^1(1!)}$ or $\! \frac{5!}{5}$	even; yes; yes	5	$\operatorname{lcm} \{ 5 \}$
Total (7 rows, 7 being the number of unordered integer partitions of 5)	--	--	--	120 (equals order of the group)	--	odd: 60 (3 classes) even;no: 36 (3 classes) even;yes;yes: 24 (1 class)	[SHOW MORE] order 1: 1 (1 class) order 2: 25 (2 classes) order 3: 20 (1 class) order 4: 30 (1 class) order 5: 24 (1 class) order 6: 20 (1 class)