In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures.
The exponential formula is a power series version of a special case of Faà di Bruno's formula.
Here is a purely algebraic statement, as a first introduction to the combinatorial use of the formula.
For any formal power series of the form
exp f ( x ) =
π =
π
runs through all partitions
of the set
the product is empty and by definition equals
One can write the formula in the following form:
exp
th complete Bell polynomial.
Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:
exp
stands for the cycle index polynomial for the symmetric group
, defined as:
σ ∈
denotes the number of cycles of
of size
This is a consequence of the general relation between
and Bell polynomials:
In combinatorial applications, the numbers
count the number of some sort of "connected" structure on an
-point set, and the numbers
count the number of (possibly disconnected) structures.
The numbers
count the number of isomorphism classes of structures on
points, with each structure being weighted by the reciprocal of its automorphism group, and the numbers
count isomorphism classes of connected structures in the same way.