is one partition of 
.
How many ways can a structure of a certain type be imposed? Let 
be the exponential generating function for a structure of type M.
Then there are 
ways that an n-set can be structured according to M. For
example, there is one way that the uniform structure can be imposed on
any non-empty n-set. Therefore, the generating function for the
uniform structure is 
.
Now consider the notion of structuring a set of structures. In how many
ways can this be done? In general, if there are several structures of type 
,
and the structures are themselves organized into a structure of type 
then the generating function for
structured
structures is 
.
: Structure Composition or Structure Substitution.
Let 
be the generating function for a structure of type ,
and similarly for 
.
The number of ways that an n-set can be first partitioned into subsets,
then each subset structured by ,
and then these structured subsets taken as objects and structured by
is given by the coefficient of 
in .
Proof: Take any partition of the n set. There are 
subsets of size 1, 
subsets of size 2, 
, 
subsets of size k. Thus 
.
We may permute subsets of the same cardinality, and within any subset we
may permute the objects. Thus there are
such partitions. For each of these partitions, each subset of the
partition is structured by ,
and the number of ways that the partitioning and
structuring may be done is then
Now, the number of ways these structured subsets may be structured
by
is given by 
times this, where 
,
and the contribution to the coefficient of 
in the generating function for the compound structure is found by multiplying
by 
.
This contribution is found in
from the term 
as
Conversely, every such term represents some partition of the n-set
into q subsets with structuring of the set of subset objects according
to
(the factor )
and with structuring of each subset according to .
QED.
Now we are ready to prove the various forms of the partition theorem which have been found useful in many-body physics.
First, there are two observations that need to be made that make
the preceding theorem more general: 1. It is possible to generalize the
theorem above to the notion of weighted objects. The weight of a
single object is assigned then by some function (
above) of the single object, and need not be an integer as might have been
assumed in the proof of the structure composition theorem. As before let
the weight of a compound object be the product of the weights of the objects
making up the compound object. 2. It is possible to label any set in any
manner desired, and to each labeled set associate a distinct cluster
(some abstract mapping for now, later a cluster is defined as a connected
graph) dependent only upon the labels that the set has associated with
it (independent of any ordering that might have been utilized). The labeling
set may be of any nonempty size, even though only as many labels appear
(perhaps not distinct) as there are elements in the set. The object that
the set of a given size then represents is then the set of all clusters
of that size, and its weight is then the sum of the weights of the clusters.
These observations will be used in deriving the cumulant and Ursell expansions.
Define a cluster as any labeled subset. Define a graph as any set of clusters. Let the weight of a set of clusters be the sum of the weights of the clusters in the set. Let the weight of any graph be the product of the weights of the clusters in the graph. Let the weight of a set of graphs be the sum of the weights of the graphs. Let the number of vertices of a cluster be the number of objects in the cluster. Let the number of vertices of the graph be the sum of the numbers of vertices of the clusters in the graph. The structure composition theorem leads directly to the Linked Cluster Theorem.
. Linked Cluster Theorem. Let 
represent the set of all graphs on k vertices. Let 
be the set of all clusters on k vertices. Let the weight of any set of
graphs be the sum of the weights of the graphs in the set, and let the
weight of any graph be the product of the weights of the clusters in the
graph. Represent the weight of a graph g as 
.
Let the weight of the empty graph be 0. Then
Proof: The coefficient of 
on the left hand side may be taken as the sum of the weights of all graphs
on N vertices, so the left hand side is the generating function
for the sum of the weights of all graphs on N vertices. The generating
function of the uniform structure is .
By the structure composition theorem the generating function for the compound
structure consisting of the uniform structure of
structures is 
.
Let 
be the weight associated with a subset of size k. Make the identification
indicated above the theorem of labeled subset and cluster. Then
is the weight of the set of all clusters of size k. The function 
is then the generating function 
just defined. Make the identification of uniform structure of clusters
and graph to find that the right side is immediately another expression
for the generating function given by the left side. QED.
We may rewrite the linked cluster theorem in symbols more easily
read directly. Let G(N) be any graph on N vertices, 
be any connected graph on N vertices and W(G) be the
weight of any graph G. We then have that the linked cluster theorem
is
The weights of clusters may be any desired functions, and the rule
stating that the weight of a graph made up of clusters is the product of
the weights of the clusters immediately yields the Ursell development of
3.5
with sum of the weights of the graphs on vertices, the W(G(N))
here, being the 
there. A proper choice for the weights of the clusters immediately leads
to the cumulant expansion theorem we have already discussed, as is proven
next.
: Cumulant expansion theorem. See section 3.3
: Let the labels of the vertices in equation 3.20
range over a set 
,
and set x=1 in that equation. Let 
be any labeling of an m-vertex cluster consisting of 
labels 1, , 
k labels k. Define 
by 
,
where 
is an argument consisting of 
's,
etc. Since an m vertex cluster may be labeled with
1's, ,
k's in 
ways we find that the expression 
of equation 3.20,
is 
.
Now, with the choice of the u given after equation 3.5,
the right side of equation 3.20
becomes the right side of equation 3.5,
and similarly for the left sides of these equations, thus showing that
the cumulant expansion theorem follows as a special case of the linked
cluster theorem, with the weights of the clusters given by special sums
of products of averages of the variables that the labels of the cluster
vertices represent. QED.
To summarize, in this section we have shown that the cumulant expansion, and the various cluster theorems are simply re-namings of a theorem about labeled partitions.