Let 
be the set of all containers of k variables from a set of variables X. Let 
be the set of all possible partitions of N variables from X. Let any element of be called a graph. Each graph in is then a set of containers 
with the sum of the m's being N. (In the graphical picture, the 
are called clusters.) To each associate a function of the variables it contains, the Ursell function 
. Now, consider the graphs 
.
For each graph 
form the product of the functions that it represents, 
, and call this the weight of the graph . Now, sum over the graphs, and define this sum as 
.
Due to the fact of the partitioning used to create W, the Ursell development of section 3.5 immediately holds.
Examples of the decomposition above occur in the Mayer f-function expansion where each edge connecting two points of the graph represents a factor in a product representing the function [69], and in quantum field theory calculations using Feynman diagrams [55]. The key thing to note is that the Mayer f-function expansion is fundamentally a theorem about labeled partitions, rather than graphs. Similarly, the cumulant expansion is fundamentally a theorem about labeled partitions. In the next section we make explicit this fundamental idea, and rigorously show the connections between the counting of labeled partitions, the partition theorem, and the linked cluster theorem.