Permutations by cycle type

The conjugacy classes of the symmetric group S_n are defined by the permutations' cycle types.
The cycle types 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).

Refined rencontres numbers

The irregular triangle A181897 shows how many permutations of n=0..8 elements have cycle type i=0..21,
where i is an index number of A194602, denoting an integer partition. (E.g. column 11 counts the number of permutations with a 3-cycle and two 2-cycles.)
The columns are grouped together based on how many elements k=0..8 are moved. (There is no column for k=1, because no permutation can move only one element.)

k	0			1		2			3			4				5				6						7						8
n		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		n!
0	1	1	1																																						1
1	1	1	1	1	0																																				1
2	1	1	1	2	0	1	1	1																																	2
3	1	1	1	3	0	3	3	3	1	2	2																														6
4	1	1	1	4	0	6	6	6	4	8	8	1	3	6	9																										24
5	1	1	1	5	0	10	10	10	10	20	20	5	15	30	45	1	20	24	44																						120
6	1	1	1	6	0	15	15	15	20	40	40	15	45	90	135	6	120	144	264	1	15	90	40	120	265																720
7	1	1	1	7	0	21	21	21	35	70	70	35	105	210	315	21	420	504	924	7	105	630	280	840	1855	1	210	504	420	720	1854										5040
8	1	1	1	8	0	28	28	28	56	112	112	70	210	420	630	56	1120	1344	2464	28	420	2520	1120	3360	7420	8	1680	4032	3360	5760	14832	1	105	1260	1120	3360	2688	1260	5040	14833	40320

An entry in row n and column group k is the product of the binomial ${\tbinom {n}{k}}$ and the top entry of the column.
The sequence of top entries could be called the refined subfactorials, and might be denoted $?i$ .
So the triangle entries can be derived from Pascal's triangle and the refined subfactorials: (k implicitly derived from i.)

$T(n,i)~=~{\binom {n}{k}}~\cdot ~?i$

The last column in column group k counts the permutations with a complete k-cycle.
An interesting fact is that the refined subfactorials in these columns are factorials, namely $(k-1)!$ .
This means that among the $!k$ derangements of k elements there are $(k-1)!$ with complete k-cycles. (E.g. here are the 24 derangements with 5-cycles.)
By excluding the full cycles one could define this rather silly sequence: 0, 0, 0, 0, 3, 20, 145, 1134, 9793...

Some rows have repeating entries, e.g. row 4 contains entry 6 twice. The number of distinct entries per row is A073906.

This triangle (with the same order of partitions) is shown in Sayma (ya da Kombinasyon Hesapları) by Ali Nesin (p. 151).

Rencontres numbers

This is the triangle A098825 of rencontres numbers ordered by number of moved elements k. (Usually they are ordered by number of fixed points.)
Its entries are the product of binomials and subfactorials. The subfactorial $!k$ is the number of derangements of k elements.

$T(n,k)~=~{\binom {n}{k}}~\cdot ~!k$

k n	0	1	2	3	4	5	6	7	8
0	1
1	1	0
2	1	0	1
3	1	0	3	2
4	1	0	6	8	9
5	1	0	10	20	45	44
6	1	0	15	40	135	264	265
7	1	0	21	70	315	924	1855	1854
8	1	0	28	112	630	2464	7420	14832	14833

k n	0	1	2	3	4	5	6	7	8
0	1⋅1
1	1⋅1	1⋅0
2	1⋅1	2⋅0	1⋅1
3	1⋅1	3⋅0	3⋅1	1⋅2
4	1⋅1	4⋅0	6⋅1	4⋅2	1⋅9
5	1⋅1	5⋅0	10⋅1	10⋅2	5⋅9	1⋅44
6	1⋅1	6⋅0	15⋅1	20⋅2	15⋅9	6⋅44	1⋅265
7	1⋅1	7⋅0	21⋅1	35⋅2	35⋅9	21⋅44	7⋅265	1⋅1854
8	1⋅1	8⋅0	28⋅1	56⋅2	70⋅9	56⋅44	28⋅265	8⋅1854	1⋅14833