In
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the exponential formula (called the polymer expansion in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
) 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
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...
.
Statement
For any
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
of the form
we have
where
and the index
runs through the list of all
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of the set
. (When
the product is
empty
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
and by definition equals
.)
One can write the formula in the following form:
and thus
where
is the
th complete
Bell polynomial
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno' ...
.
Alternatively, the exponential formula can also be written using the
cycle index
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
of the symmetric group, as follows:
where
stands for the cycle index polynomial, for the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
defined as:
and
denotes the number of cycles of
of size
. This is a consequence of the general relation between
and
Bell polynomials:
Examples
*
because there is one partition of the set
that has a single block of size
, there are three partitions of
that split it into a block of size
and a block of size
, and there is one partition of
that splits it into three blocks of size
. This also follows from
, since one can write the group
as
, using cyclic notation for permutations.
* If
is the number of graphs whose vertices are a given
-point set, then
is the number of connected graphs whose vertices are a given
-point set.
* There are numerous variations of the previous example where the graph has certain properties: for example, if
counts graphs without cycles, then
counts trees (connected graphs without cycles).
* If
counts directed graphs whose (rather than vertices) are a given
point set, then
counts connected directed graphs with this edge set.
Applications
In applications, the numbers
often 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.
In quantum field theory and statistical mechanics, the
partition functions
, or more generally
correlation functions, are given by a formal sum over
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s. The exponential formula shows that
can be written as a sum over connected Feynman diagrams, in terms of
connected correlation functions.
See also
*
References
* {{Citation , authorlink=Richard P. Stanley , last1=Stanley , first1=Richard P. , title=Enumerative combinatorics. Vol. 2 , url=http://www-math.mit.edu/~rstan/ec/ , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-56069-6 , id={{ISBN, 978-0-521-78987-5 , mr=1676282 , year=1999 , volume=62 Chapter 5 page 3
Exponentials
Enumerative combinatorics