Permutations by cycle type

The conjugacy classes of the symmetric group S_n are defined by the permutations' cycle types,
which correspond to the integer partitions of n. So the number of conjugacy classes of S_n is  A000041(n).

The first 8! = 40320 finite permutations have  A000041(8) = 22 different cycle types corresponding to the 22 first integer partitions.
To determine the cycle type of a permutation of up to 8 elements see this table (a supporting file of  A198380).

The following table ( A181897) shows how many permutations of n elements have cycle type k. (Compare this table.)
Blue numbers are factorials.
The number of distinct entries per row is 1,1,3,4,6,7,11,16... =  A073906.
The difference tables always show the differences between the rows of the table above. On the left of the difference tables are always constant columns.

These are refined rencontres numbers.

This table ( A181897, black) is a refinement of the rencontres numbers ( A008290, red) - it shows the same information more detailed. How many permutations of 5 elements change 4 elements? Rencontres numbers: rn(5,1) = 45         (5-4 = 1 is the number of fixed points.) How many permutations of 5 elements have two 2-cycles? Refined rencontres numbers: rrn(5,3) = 15 How many permutations of 5 elements have have a 4-cycle? Refined rencontres numbers: rrn(5,4) = 30

	0	1 2	2 3	3 2,2	4 4	5 3,2	6 5	7 2,2,2	8 4,2	9 3,3	10 6	11 3,2,2	12 5,2	13 4,3	14 7	15 2,2,2,2	16 4,2,2	17 3,3,2	18 6,2	19 5,3	20 4,4	21 8	Σ
1	1																						1! = 1
2	1	1																					2! = 2
3	1	3	2																				3! = 6
4	1	6	8	3	6																		4! = 24
5	1	10	20	15	30	20	24																5! = 120
6	1	15	40	45	90	120	144	15	90	40	120												6! = 720
7	1	21	70	105	210	420	504	105	630	280	840	210	504	420	720								7! = 5040
8	1	28	112	210	420	1120	1344	420	2520	1120	3360	1680	4032	3360	5760	105	1260	1120	3360	2688	1260	5040	8! = 40320
Differences 1
2		1																					1
3		2	2																				4
4		3	6	3	6																		18
5		4	12	12	24	20	24																96
6		5	20	30	60	100	120	15	90	40	120												600
7		6	30	60	120	300	360	90	540	240	720	210	504	420	720								4320
8		7	42	105	210	700	840	315	1890	840	2520	1470	3528	2940	5040	105	1260	1120	3360	2688	1260	5040	35280
Differences 2
3		1	2																				3
4		1	4	3	6																		14
5		1	6	9	18	20	24																78
6		1	8	18	36	80	96	15	90	40	120												504
7		1	10	30	60	200	240	75	450	200	600	210	504	420	720								3720
8		1	12	45	90	400	480	225	1350	600	1800	1260	3024	2520	4320	105	1260	1120	3360	2688	1260	5040	30960
Differences 3
4			2	3	6																		11
5			2	6	12	20	24																64
6			2	9	18	60	72	15	90	40	120												426
7			2	12	24	120	144	60	360	160	480	210	504	420	720								3216
8			2	15	30	200	240	150	900	400	1200	1050	2520	2100	3600	105	1260	1120	3360	2688	1260	5040	27240
Differences 4
5				3	6	20	24																53
6				3	6	40	48	15	90	40	120												362
7				3	6	60	72	45	270	120	360	210	504	420	720								2790
8				3	6	80	96	90	540	240	720	840	2016	1680	2880	105	1260	1120	3360	2688	1260	5040	24024

Tables

Array of 2-element subsets of an 8-element set
(compare  A018900 as a square array)

All tables of 8-element permutations for this topic are organized in the same way:

• The numbers in the leftmost column denote only the rows of the table.
• In column # are the unique index numbers of the finite permutations (compare  A195663).
• The next column contains permutations of 0...7 and 1...8 without spaces in small script, to make the table searchable (Ctrl+F). Sorting by this column brings the permutations in lex order, while the default order is reverse colex.
• The next eight columns show the actual permutations. Elements out of their natural order are in bold case.
• Column PM shows the permutation matrices. (Code can be copied from these fields.)
• The next seven columns show the inversion vector without its first element (which is always zero).
• Column IS shows which 2-cycles are part of the inversion set. Compare file on the right.
• Column IN shows the inversion numbers. The inversion number is the sum of the inversion vector and also the cardinality of the inversion set.
• The next column is a shorthand of those on the right without spaces in small script. These seven numbers tell how many red squares are in the columns of the arrays in IS.

To sort by more than one column you can hold the shift-key and than click the other sort buttons. *

In the default order (ordered by #) the background color of the rows changes when a new element is out of it's natural order. (Or, in other terms, when a new place on the right is necesseary to write the permutation.)

Only the table of transpositions is shown on this page.
The tables of permutations with other cycle types have own pages in this category.

Permutations with cycle type 1 (one 2-cycle)

The transpositions ordered in an array

The inversion sets in column IS show much regularity:

• In the table's default order sets with the same red square on the lowest position are grouped together (their rows have the same background color), and the size of the red triangle increases from top to bottom.
• When the table is first ordered by the inversion numbers in column IN and then by #, sets with the same size of the red triangle are grouped together. Within these sections the position sinks from top to bottom. From section to section the triangle size increases.

	#										PM								IS	IN
1	1	21345678 10234567	2	1	3	4	5	6	7	8		1	0	0	0	0	0	0		1	1000000	1	0	0	0	0	0	0
2	2	13245678 02134567	1	3	2	4	5	6	7	8		0	1	0	0	0	0	0		1	1000000	1	0	0	0	0	0	0
3	5	32145678 21034567	3	2	1	4	5	6	7	8		1	2	0	0	0	0	0		3	2100000	2	1	0	0	0	0	0
4	6	12435678 01324567	1	2	4	3	5	6	7	8		0	0	1	0	0	0	0		1	1000000	1	0	0	0	0	0	0
5	14	14325678 03214567	1	4	3	2	5	6	7	8		0	1	2	0	0	0	0		3	2100000	2	1	0	0	0	0	0
6	21	42315678 31204567	4	2	3	1	5	6	7	8		1	1	3	0	0	0	0		5	2210000	2	2	1	0	0	0	0
7	24	12354678 01243567	1	2	3	5	4	6	7	8		0	0	0	1	0	0	0		1	1000000	1	0	0	0	0	0	0
8	54	12543678 01432567	1	2	5	4	3	6	7	8		0	0	1	2	0	0	0		3	2100000	2	1	0	0	0	0	0
9	80	15342678 04231567	1	5	3	4	2	6	7	8		0	1	1	3	0	0	0		5	2210000	2	2	1	0	0	0	0
10	105	52341678 41230567	5	2	3	4	1	6	7	8		1	1	1	4	0	0	0		7	2221000	2	2	2	1	0	0	0
11	120	12346578 01235467	1	2	3	4	6	5	7	8		0	0	0	0	1	0	0		1	1000000	1	0	0	0	0	0	0
12	264	12365478 01254367	1	2	3	6	5	4	7	8		0	0	0	1	2	0	0		3	2100000	2	1	0	0	0	0	0
13	390	12645378 01534267	1	2	6	4	5	3	7	8		0	0	1	1	3	0	0		5	2210000	2	2	1	0	0	0	0
14	512	16345278 05234167	1	6	3	4	5	2	7	8		0	1	1	1	4	0	0		7	2221000	2	2	2	1	0	0	0
15	633	62345178 51234067	6	2	3	4	5	1	7	8		1	1	1	1	5	0	0		9	2222100	2	2	2	2	1	0	0
16	720	12345768 01234657	1	2	3	4	5	7	6	8		0	0	0	0	0	1	0		1	1000000	1	0	0	0	0	0	0
17	1560	12347658 01236547	1	2	3	4	7	6	5	8		0	0	0	0	1	2	0		3	2100000	2	1	0	0	0	0	0
18	2304	12375648 01264537	1	2	3	7	5	6	4	8		0	0	0	1	1	3	0		5	2210000	2	2	1	0	0	0	0
19	3030	12745638 01634527	1	2	7	4	5	6	3	8		0	0	1	1	1	4	0		7	2221000	2	2	2	1	0	0	0
20	3752	17345628 06234517	1	7	3	4	5	6	2	8		0	1	1	1	1	5	0		9	2222100	2	2	2	2	1	0	0
21	4473	72345618 61234507	7	2	3	4	5	6	1	8		1	1	1	1	1	6	0		11	2222210	2	2	2	2	2	1	0
22	5040	12345687 01234576	1	2	3	4	5	6	8	7		0	0	0	0	0	0	1		1	1000000	1	0	0	0	0	0	0
23	10800	12345876 01234765	1	2	3	4	5	8	7	6		0	0	0	0	0	1	2		3	2100000	2	1	0	0	0	0	0
24	15960	12348675 01237564	1	2	3	4	8	6	7	5		0	0	0	0	1	1	3		5	2210000	2	2	1	0	0	0	0
25	21024	12385674 01274563	1	2	3	8	5	6	7	4		0	0	0	1	1	1	4		7	2221000	2	2	2	1	0	0	0
26	26070	12845673 01734562	1	2	8	4	5	6	7	3		0	0	1	1	1	1	5		9	2222100	2	2	2	2	1	0	0
27	31112	18345672 07234561	1	8	3	4	5	6	7	2		0	1	1	1	1	1	6		11	2222210	2	2	2	2	2	1	0
28	36153	82345671 71234560	8	2	3	4	5	6	7	1		1	1	1	1	1	1	7		13	2222221	2	2	2	2	2	2	1