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 Template:Oeis(n).

The first 8! = 40320 finite permutations have Template:Oeis(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 Template:Oeis).

Refined rencontres numbers

The irregular triangle Template:Oeis shows how many permutations of n=0..8 elements have cycle type i=0..21,
where i is an index number of Template:Oeis, 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.)

An entry in row n and column group k is the product of the Template:W $(\genfrac{}{}{0ex}{}{n}{k})$ 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 Template:W and the refined subfactorials:   (k implicitly derived from i.)

$T(n,i)=(\genfrac{}{}{0ex}{}{n}{k})\cdot ?i$

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 Template:Oeis.

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 Template:Oeis of Template:W 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 Template:Ws of k elements.

$T(n,k)=(\genfrac{}{}{0ex}{}{n}{k})\cdot !k$