A numerical example is provided by . The set of proper partitions characterizing the conjugacy classes of is
the set of corresponding cycle types is
Thus the types of turn out to be
The orders of the conjugacy classes are 6,18,12,1,9,4,6,4,12. We now describe an interesting action of which is in fact an action of the form .
. Example The action of on is obviously similar to the following action of on the set :
The corresponding permutation group on will be denoted by
and called the composition of and , while
will be used for the permutation group on , induced by the natural action of on .
The action of the wreath product on induces a natural action of on the set
i.e. on the set of 0-1-matrices consisting of rows and columns:
Since , we can do this in two steps:
Hence we can first of all permute the columns of in such a way that the numbers of 1's in the columns of the resulting matrix is nonincreasing from left to right: And after having carried out this permutation with a suitable , we can find a that moves the 1's of each column in flush top position. This proves that the orbit of under is characterized by an element of the form
i.e. by a proper partition of . Hence the orbits of on are characterized by the proper partitions , where each part and where the total number of parts is :
. Corollary
There exists a natural bijection
Hence an application of the Cauchy-Frobenius Lemma yields the following formula for the number of partitions of this form:
which can be made more explicit by an application of .