Suppose we are given a permutation group 
on a set 
. It induces
interesting structures both on and on that are closely related (for
more details cf. [7]). To begin with, there are the orbits of the elements 
: 
, two of which are either identical or disjoint. Therefore, the
set
of all the orbits is a set-partition of . To the orbits there
correspond the stabilizers (which are in fact subgroups): 
, and the close relationship between orbits and
stabilizers is the existence of the following natural bijection between an
orbit of an element and the set of left cosets of its stabilizer:
The cycle index of is the following polynomial 
in the
indeterminates 
over 
, defined by
where 
is the cycle type of the permutation
. This means, 
decomposes into 
disjoint cycles of length
for 
. All elements of a conjugacy class have the same
cycle type, so the cycle index can be computed in the following way:
where 
is the system of conjugacy classes of and where 
is a
representative of the conjugacy class 
.
Let and 
denote permutation groups on and 
, respectively. The
wreath product 
is a permutation group on the set
, defined by
with multiplication 
, where
and 
.
In the case when 
and 
we write 
instead of 
.
The wreath product acts in a natural way on 
. The effect of
the permutation 
on 
is
The following lemma ([9][10]) reduces the action of a wreath
product to the action of the group on the set of all functions from
into the set of all orbits of on :
Moreover, the mapping
is a bijection if 
.
Lehmann's Lemma:
If and denote permutation groups on and , respectively,
then acts on 
in the following way:
is given by 
.
There are many enumerative and constructive results dealing with various
group actions on the set induced by permutation groups on the domain
and on the range . For example, 
acts upon 
by
The corresponding generating function for the numbers of -orbits on is given by (see [1]):
The group action of above can be restricted to the set of all injective
mappings from 
to , the corresponding generating function is
