Examples

### Examples

A numerical example is provided by S 3 wr S 2. The set of proper partitions characterizing the conjugacy classes of S 2 is
{ a | a2 }= {(2),(12) },
the set of corresponding cycle types is
{a | a |¾| 2 }= {(0,1),(2,0) }.
Thus the types of S 3 wr S 2 turn out to be
 0 1 0 0 0 0
 0 0 0 1 0 0
 0 0 0 0 0 1
 2 0 0 0 0 0
 0 0 2 0 0 0
 0 0 0 0 2 0
 1 0 1 0 0 0
 1 0 0 0 1 0
 0 0 1 0 1 0 .
The orders of the conjugacy classes are 6,18,12,1,9,4,6,4,12. We now describe an interesting action of S m wr S n which is in fact an action of the form GYX.
Example: The action of S m wr S n on mn is obviously similar to the following action of S m wr S n on the set m´n:
S m wr S n ´( m´n) -> m´n:(( y, p),(i,j)) -> ( y( pj)i, pj).
The corresponding permutation group on m´n will be denoted by
S n [ S m ]
and called the composition of S n and S m , while
G[H]
will be used for the permutation group on Y ´X, induced by the natural action of H wr X G on Y ´X.

The action of the wreath product S m wr S n on m´n induces a natural action of S m wr S n on the set

YX:=2 m´n= {(aij) | aij Î{0,1 },i Î m,j Î n },
i.e. on the set of 0-1-matrices consisting of m rows and n columns:
S m wr S n ´2 m´n :(( y, p),(aij)) -> (a y-1(j)i, p-1 j).
Since ( y, p)=( y,1)( i, p), we can do this in two steps:
(aij) -> (ai, p-1j) -> (a y-1(j)i, p-1j).
Hence we can first of all permute the columns of (aij) in such a way that the numbers of 1's in the columns of the resulting matrix is nonincreasing from left to right: åi ai, p-11 ³åi ai, p-12 ³... And after having carried out this permutation with a suitable p, we can find a yÎS m * that moves the 1's of each column in flush top position. This proves that the orbit of (aij) under S m wr S n is characterized by an element of the form
 1 ... ... ... 1 . . 1 ... 1 0
(which is an element of 2 m´n), i.e. by a proper partition of k:= åi,jaij. Hence the orbits of S m wr S n on 2 m´n are characterized by the proper partitions a, where each part ai £n and where the total number of parts is £m:
Corollary: There exists a natural bijection
S m wr S n \\2 m´n -> { ak | k <= mn, a1 <= n, l( a) £m }.
Hence an application of the Cauchy-Frobenius Lemma yields the following formula for the number of partitions of this form:
| S m wr S n \\2 m´n | = (m!nn!)-1 å( y, p) ÎS m wr S n 2 S n c(h n( y, p)),
which can be made more explicit by an application of the Lemma.

harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001

 Examples