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.

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 