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
Y^{X}:=2^{ m´n}= {(a_{ij})  a_{ij} Î{0,1 },i Î m,j Î n },
i.e. on the set of 01matrices consisting of m rows and n columns:
S_{ m} wr S_{ n} ´2^{ m´n} :(( y, p),(a_{ij})) > (a_{ y1(j)i, p1 j}).
Since ( y, p)=( y,1)( i, p),
we can do this in two steps:
(a_{ij}) > (a_{i, p1j})
> (a_{ y1(j)i, p1j}).
Hence we can first of all permute the columns of (a_{ij}) in
such a way that the numbers of 1's in the columns of the resulting matrix
is nonincreasing from left to right:
å_{i} a_{i, p11} ³å_{i} a_{i, p12} 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 (a_{ij})
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:= å_{i,j}a_{ij}. Hence the orbits of
S_{ m} wr S_{ n} on 2^{ m´n} are characterized by the
proper partitions a, where each part a_{i} £n and where the total number of parts is £m:
Corollary:
There exists a natural bijection
S_{ m} wr S_{ n} \\2^{ m´n} > { a¾k  k <= mn, a_{1} <= n, l( a) £m }.
Hence an application of the CauchyFrobenius Lemma yields
the following formula for the number of partitions of this form:
 S_{ m} wr S_{ n} \\2^{ m´n}  =
(m!^{n}n!)^{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.