\documentclass[12pt]{amsart}
%third revised version submitted to Seminaire Lotharingien de Combinatoire May 30, 2011
\title{Combinatorial Species and Feynman Diagrams}

\author{William G. Faris}

\address{Department of Mathematics\\University of Arizona\\Tucson, AZ 85711}
\email{faris@math.arizona.edu}
\date{}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\newenvironment{example}{Example: }{}

\newcommand{\Rb}{\mathbf{R}}
\newcommand{\Fc}{\mathcal{F}}
\newcommand{\Pc}{\mathcal{P}}
\newcommand{\Xc}{\mathcal{X}}
\newcommand{\Yc}{\mathcal{Y}}
\newcommand{\Feyn}{\mathrm{Feyn}}
\newcommand{\Par}{\mathrm{Par}}
\newcommand{\Mat}{\mathrm{Mat}}
\newcommand{\dom}{\mathrm{dom}}

\begin{document}

\dedicatory{
Dedicated to the memory of Pierre Leroux
}

\begin{abstract}

A Feynman diagram is a graphical construction
that describes certain interactions in physics.
Most calculations with such diagrams reduce to consideration of connected
Feynman diagrams. These in turn may be constructed from
line-irreducible Feynman diagrams, those for which removal
of a single line does not destroy connectivity. The purpose of
this article is to exhibit the combinatorial nature of
this construction in the framework of species of structures.
The main result is a dissymmetry theorem for connected Feynman diagrams.
This purely combinatorial theorem relates the species of connected diagrams to
species with less symmetry, such as the species of connected
diagrams with a designated line-irreducible subdiagram.
There is also a discussion of the relation of this result to the
Legendre transform.


\end{abstract}

\maketitle

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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 61A \rms (2011), Article~B61An\hfill}
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\section{Introduction}

This article is on the relation between constructions in
combinatorics and corresponding constructions in physics.
The framework is the theory of species of
structures in the sense of Joyal \cite{J} (see also the book \cite{BLL}). The physics topic is Feynman
diagrams. Feynman diagrams are combinatorial structures
involving graphs constructed in a rather complicated way.

In graph theory there is a fundamental concept of connected graph.
There are also stronger notions of connectivity. A connected graph is
2-connected if it does not have a cut-vertex: a vertex whose removal disconnects the graph. A connected
graph is 2-edge-connected if it does not have a bridge: an edge whose removal disconnects the graph.
In the Feynman diagram context the notion corresponding to 2-edge-connected graph is 1-particle-irreducible
diagram. For simplicity we henceforth use the terminology line-irreducible diagram.


There are identities that reduce
sums over connected diagrams to sums over line-irreducible
diagrams. They may be formulated in terms of the
Legendre transform. Such results are standard in the physics literature
and may be found in textbooks \cite[Chapter~16]{W}.
Current discussions may be found, for instance, in \cite{dlPS} or \cite{JKM}.
The lecture notes \cite{E} are a particularly  good introduction.
The goal of the present work is to formulate and prove these identities in
the context of species of structures. The identities should take the form of
bijections between sets of combinatorial structures.  In short, the numerical identities
should be  consequences of combinatorial identities.

This species approach  depends on a combinatorial fact about trees
known as the dissymmetry theorem. This principle was applied to statistical mechanics
by Leroux \cite{L}. In his treatment the natural setting is graphs
with the concept of 2-connectivity.  An article by Brydges and Leroux \cite{BL} explored what happens in
the graph case with the 2-edge-connectivity concept. For the Feynman diagram case considered
here the natural notion is line-irreducibility.

The species identities are  relations between finite sets. When infinite series
are encountered, they are interpreted in a formal sense. See \cite{F} for a survey of species
techniques and of their applications to statistical mechanics.  The formal combinatorics
of Feynman diagrams (with oriented lines) is systematically explored by
Abdesselam in \cite{A}.
Another topic of analysis  is concerned with
convergence of expansions. See, for example, Brydges and Martin \cite[Section~VIII]{BM}
for the state of the art.

The Feynman diagrams under consideration here are labeled Feynman diagrams.
The theory of combinatorial species works particularly smoothly with labeled structures. In this case the only symmetry
considered is permutation of the labels, and the corresponding generating series (exponential generating function)
is a relatively uncomplicated construct. That is the reason for
concentrating on that case in the present treatment.
By contrast, many expositions deal with isomorphism classes of Feynman diagrams. These are
abstract graphs with vertices and edges, allowing multiple edges and loops. The reason for
using isomorphism classes of diagrams is to compress the sums involved so as to have less
terms to compute. This is important in physics for practitioners of perturbation theory who need
to evaluate some quantity---say a scattering amplitude---up to some order in the coupling constant.
The theory of combinatorial species also describes isomorphism classes, but
this requires a deeper understanding of the automorphisms that describe the symmetries. The
natural tool is then the cycle index series.

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\markboth{\SMALL WILLIAM G. FARIS}{\SMALL 
COMBINATORIAL SPECIES AND FEYNMAN DIAGRAMS}
%
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While the isomorphism class description of Feynman diagrams is not needed
for the main body of the present article, it is relevant to understanding
its relation to other approaches to the combinatorics of Feynman diagrams.
For this reason, an appendix is included that discusses this topic. In particular, this
allows a more detailed comparison with the contributions \cite{BL} and \cite{A}.




Feynman diagrams have an
important interpretation in quantum field theory. They describe the
perturbation of free quantum particle propagation by interactions.
The vertices represent the interactions, and the lines represent
the free particle propagation. Feynman diagrams also have an interpretation
in the theory of spatial stochastic processes. Here they describe the
perturbation of a Gaussian process by nonlinear interactions. The vertices
come from the interaction terms, and the lines represent the Gaussian
covariances. In the following we shall use the language of perturbations
of Gaussian processes. This corresponds  to the quantum field
case with chargeless bosons. The lack of charge corresponds to the
combinatorial fact that the lines are unoriented.




\section{Feynman diagrams}

\subsection{Exponential generating functions}

For each $n$ consider a finite set $U$ with
cardinality $n$. A point $i$ in $U$ is called a \emph{label}.
One possible choice is $U = \{1,\ldots, n\}$. However, it can be useful
to employ a subset $V$ of $U$ as another label set. In other circumstances
one can take a set partition $\Gamma$ of $U$ as a label set. So it is best
to allow a general finite set at the outset. Combinatorial objects are often built from  such a label set; the exact
nature of this set is usually not important.

 Fix a  finite set $\Xc$ (possibly very large). This is the color palette.
Each $x$ in $\Xc$ is a \emph{color}.
Consider a function $a: U \to \Xc$. This is
called a \emph{colored set}. Thus the label $i$ has color $a_i$.
The case
where we do not care to use colors is obtained by using a color palette with only
one color, for instance, gray. We may refer to this situation as \emph{monochromatic}.

In some interpretations the colors may be thought of as the index set for a
stochastic process. Each color is a point in space (or a point in time).
A colored set is a way of describing several points. Thus, for instance,
for a colored set $a: \{i, j\} \to \Xc$ this describes
points $a_i, a_j$ in space, which may be the same point.

Such an index set may  also be thought of as the coordinate index set for a product space of dimension $N$,
where $N$ is the number of colors. An element $\phi$ of this product space is an $N$-component vector
that assigns to each index $x$ in $\Xc$ a number $\phi(x)$, the $x$ component of the vector.
More generally, a tensor of rank 2 assigns to each ordered pair of indices $x,y$ a number $\phi(x,y)$.
A tensor of rank $n$ is defined in a similar way as a number depending on $n$ indices.
In the applications considered here the tensors are symmetric tensors. Thus in the rank 2 case
we have $\phi(x,y) = \phi(y,x)$.

The notion of colored set gives a more flexible way of describing such tensors.
Consider a label set $U_n$ of cardinality $n$.
Say that for each colored set $a: U_n \to \Xc$ there
is a corresponding coefficient $f(a)$. Suppose that $f(a)$ depends only
on the number of points in $U_n$ of each color. Then  $f(a)$ is a
\emph{symmetric tensor} of rank $n$ in a space of dimension $N$.
The colors are the tensor indices. For example, if $n=2$ the tensor is rank 2.
For a colored set $a: \{i,j\} \to \Xc$ there are two indices $a_i, a_j$, which
may be the same. Permuting the labels gives the same value of $f(a)$.

The \emph{exponential generating function} for this family of tensors is a formal
power series in many variables $z_x$, one for each color $x$ in $\Xc$. It is given
by the formal sum
\begin{equation}
F(z) = \sum_{n=0}^\infty \frac{1}{n!} \sum_{a: U_n \to \Xc} f(a) z^a,
\end{equation}
where $z^a = \prod_{j \in U_n} z_{a_j}$. In the monochromatic case
the tensors are scalar coefficients that depend only on $n$, and there is only one variable.

Exponential generating functions give a convenient summary of  important
tensor operations. The most fundamental operation is  combinatorial
multiplication.
Say that $G(z)$ corresponds to the tensors $g(a)$, and $H(z)$ corresponds to the
tensors $h(a)$. Then the product $F(z) = G(z) H(z)$ corresponds to the tensors
\begin{equation}
f(a) = \sum_{\langle V, W \rangle} g(a_V) h(a_W),
\end{equation}
where $V, W$ range over ordered pairs of disjoint sets with union $U$, and
where $a_V$ and $a_W$ are the restrictions of $a$ to $V$ and $W$. Combinatorial
multiplication splits the set in all possible ways, multiplies the weights
of the constituents, and adds the results.

Composition is even more important. It is useful to allow two  color
palettes. Say that $G(w)$ is defined with color palette $\Yc$. Furthermore, say
that for each $y$ in $\Yc$ there is an exponential generating function $H_y(z)$ with
color palette $\Xc$. Suppose for simplicity that each $H_y(0) = 0$. Then the composition $F(z) = G(H(z))$ makes sense as an
exponential generating function with color palette $\Xc$. The corresponding
operation on tensors $a: U \to \Xc$ is
\begin{equation}
f(a) = \sum_\Gamma \sum_{b: \Gamma \to \Yc} g(b) \prod_{V \in \Gamma} h_{b_V}(a_V),
\end{equation}
where $a: U \to \Xc$, and where $\Gamma$ ranges over set partitions of $U$. Combinatorial composition
partitions the set in all possible ways, with all possible colorings of the
blocks, multiplies the weight of the partition with the weights
of the blocks of the partition, and adds the results. If it happens that the set $U$ is
empty, then the only set partition is the empty partition $\Gamma$ with no blocks.
The product over the blocks $V$ in $\Gamma$ is the empty product with value one.

An important special case is when $G(w)$ is monochromatic and there is only one
$H(z)$. In particular, if we take the monochromatic $G(w) =\exp(w)$, then
the corresponding tensors are simply the scalar 1. So in that case $F(z) = \exp(H(z))$ is
given by
\begin{equation}
f(a) = \sum_\Gamma \prod_{V \in \Gamma} h(a_V),
\end{equation}
where $a: U \to \Xc$, and where $\Gamma$ ranges over set partitions of $U$.
The combinatorial exponential partitions the set in all possible ways, multiplies the  weights
of the blocks of the partition, and adds the results.

%The derivative has an interesting interpretation. If $F(z)$ is an exponential
%generating function with coefficients $f(a)$, then the partial derivative $ \partial F(z)/\partial z_x$
%has coefficients $f(a^x)$. Here $a: U \to \Xc$, $*$ is an external point (not in $U$), and $a^x: U + \{*\} \to \Xc$
%agrees with $a$ on $U$ and has value $x$ on $*$. In other words, the derivative gives the weight of
%the set with an additional external point of the designated color.





\subsection{Gaussian expectations}

Let $\Xc$ be a palette of colors, and consider Gaussian random variables $\phi^x$ indexed
by $x$ in $\Xc$. This means that there is a Gaussian measure on the $N$-dimensional product space $\Rb^\Xc$. The
$\phi^x$ are the coordinate variables with induced one-dimensional Gaussian measures. The expectation of
a function $h(\phi)$ with respect to the Gaussian measure will be denoted $\langle h(\phi) \rangle_0$.
A Gaussian measure is always determined by its mean function and its covariance function.
In the following suppose that the Gaussian random variables have mean zero and covariance
\begin{equation}
C^{xy} = \langle \phi^x \phi^y \rangle_0.
\end{equation}
We could also write the covariance as $C^a$, where $a: \{i,j\} \to \Xc$ is a colored set with
$a_i = x$ and $a_j = y$. In other words, it is a rank 2 tensor.

An expectation of a monomial in random variables is called a moment. In particular, a covariance
of mean zero random variables is a second moment.
The higher moments $\langle \phi^a \rangle_0 = \langle \prod_j \phi^{a_j} \rangle_0$
of the family of Gaussian random variables should be determined by the exponential generating function
\begin{equation}
\langle \exp( \sum_x J_x \phi^x) \rangle_0 = \sum_{n = 0}^\infty \frac{1}{n!} \sum_{a: U_n \to \Xc}
\langle \phi^a \rangle_0 J_a,
\end{equation}
where $J_a = \prod_j J_{a_j}$. Here the $J_a$ are arbitrary complex variables. In the Gaussian
case this expression is well-defined.
Furthermore, since the variables are Gaussian, the moment generating function has an explicit
expression in terms of the covariance:
\begin{equation}
\langle \exp( \sum_x J_x \phi^x ) \rangle_0 = \exp( \frac{1}{2} \sum_x \sum_y C^{xy} J_x J_y)=
\exp( \frac{1}{2} \sum_{a: U_2 \to \Xc} C^a J_a).
\end{equation}
It follows from the combinatorial exponential
that the higher moments are given in terms of covariances for $a: U \to \Xc$ by
\begin{equation}
\langle \phi^a \rangle_0 =  \sum_{\sigma \in \Mat[U]} \prod_{W \in \sigma} C^{a_W},
\end{equation}
where $\sigma$ ranges over all set partitions of $U$ into two-point blocks.
Such a partition is called a \emph{perfect matching} of $U$. Write the weight
of the perfect matching $\sigma$ as
\begin{equation}
C(a,\sigma) =\prod_{W \in \sigma}  C^{a_W}.
\end{equation}
Then
\begin{equation}
\langle \phi^a \rangle_0= \sum_{\sigma \in \Mat[U]}  C(a,\sigma)
\end{equation}
is the sum of weights of perfect matchings.



\subsection{Vacuum Feynman diagrams}

In the theory of random fields non-Gaussian processes  are built from Gaussian
processes. Start with a set $\Xc$ indexing a Gaussian process. Consider variables $\phi^x$ for $x \in \Xc$.
The \emph{action} written in terms of the fields $\phi$ takes the form
\begin{equation}
S(\phi) = \sum_{n=1}^\infty \frac{1}{n!} \sum_{a: U_n \to \Xc} S_{a} \phi^a.
\end{equation}
The $\phi^a$ is the product $\prod_j \phi^{a_j}$, where
the product is over the label set. The coefficients $S_a$ are symmetric tensors.
The action is a given quantity that defines the process of interest.
In some applications it is restricted to be a polynomial of low
degree, such as three or four. In that case, there are only finitely many
parameters that define the action.


The exponential of the action is
\begin{equation}
\exp(S(\phi)) = \sum_{n = 0}^\infty \frac{1}{n!} \sum_{a: U_n \to \Xc} [ \sum_{\Gamma\in \Par[U_n]}\prod_{V \in \Gamma} S_{a_V} ]  \phi^a.
\end{equation}
The $\Gamma$ range over set partitions of the label set. The $a_V$ is the restriction of $a$ to the subset $V$ of
the label set. In the future it will be convenient to write
\begin{equation}
S(a,\Gamma) = \prod_{V \in \Gamma} S_{a_V}.
\end{equation}
The  \emph{partition function} is the expectation
\begin{equation}
Z = \langle \exp( S(\phi) \rangle_0 = \sum_{n = 0}^\infty \frac{1}{n!} \sum_{a: U_n \to \Xc}
  \sum_{\Gamma\in \Par[U_n]}\sum_{\sigma \in \Mat[U_n]} S(a,\Gamma) C(a,\sigma).
\end{equation}

A \emph{vacuum Feynman diagram}  $\gamma$ for  label set $U$
consists of a set partition $\Gamma$ of
$U$ and a perfect matching $\sigma$ of $U$.
Each block in $\Gamma$ is a \emph{vertex}, and each pair $ \{i,j\}$ in $\sigma$ is a \emph{line}.
The labels $i,j$  are the ends of the lines.
The weight of the vacuum Feynman diagram is the product $\Feyn(a,\gamma) = S(a, \Gamma) C(a,\sigma)$. The result above says
that the partition function is given by a sum of weights of diagrams.

Each vacuum Feynman diagram defines a graph (with loops and multiple edges). Each block in the set partition
$\Gamma$ is a vertex, and  an edge from such a vertex to another corresponds to a line with
one end in one vertex and the other end  in the other vertex.

The partition function $Z$ may be thought of as a function of the parameters that define the
action. The physics is determined by this function. The partition function  does not
have a direct physical interpretation, but ratios such as the logarithmic differential $dZ/Z$
determine relevant expected values. The quantities that vary in such a differential expression
are certain parameters in the action. The particular parameters that are chosen depend on the
physical quantity that one is trying to compute.
In physics it is usual to focus on the \emph{connected function} or \emph{free energy} $F$  defined
so that $Z = \exp(F)$. Then the differential $dF = dZ/Z$ is a quantity of interest.
The combinatorial term  ``connected function'' comes from the fact that its computation involves connected Feynman diagrams.
In physics ``free energy'' often refers loosely to a quantity that is a multiple of a thermodynamic
free energy. In any case the quantity $F$, regarded as a function of action parameters, is central
to the theory.

\subsection{Feynman diagrams with  legs and with amputated lines}

There are more general kinds of Feynman diagrams. A \emph{Feynman diagram with legs} for label set $U$
consists of a set partition $\Gamma$ of a subset $U_0$ of $U$
together with a perfect matching $\sigma$ of $U$. A line that does not have both ends in $U_0$
is called a \emph{leg}. A label that is not in the vertex subset $U_0$ is a \emph{leg label}.
One way to get a Feynman diagram with legs is to take a vacuum Feynman diagram,
remove all the one-point vertices, and replace them with leg labels.

Another kind of Feynman diagram has vertices with labels that do not belong to lines.
An \emph{amputated Feynman diagram} $\gamma$ on a label set $U$ is a set partition $\Gamma$ of $U$
together with a perfect matching $\sigma$ of a subset of $U$. A label that is not matched is an
\emph{amputated line label}. It may be thought of as one end point of a potential or virtual line, and may
be pictured as a half-line.

The most general form of Feynman diagram $\gamma$ on label set $U$ would consist of a set partition
$\Gamma$ of a subset of $U$ and a perfect matching $\sigma$ of another subset of $U$. Then there could be ordinary
labels that belong both to the partition and the perfect matching, leg labels that belong
to the perfect matching but not the partition, amputated line labels that belong to
the partition but not the perfect matching, and amputated leg labels that belong to neither one.
In each case the weight of the diagram is $\Feyn(a,\gamma) = S(a,\Gamma) C(a,\sigma)$.

\section{Species of structures}

\subsection{Species equations}

The most basic species notion is that of an assignment to each finite set $U$ of a finite set $F[U]$ of combinatorial objects,
with a suitable naturality requirement. Often $U$ is called the label set, and the elements of $F[U]$
are constructed from $U$. An elementary example is the power set species $P$ that assigns to each
set $U$ with $n$ elements the set $P[U]$ of all subsets of $U$, which is a set with $2^n$ elements.
Many other examples appear in \cite[Chapter~1]{BL}.

There is a more general concept of weighted species. A weighted species is one for which counting of $F[U]$ is replaced by adding weights associated to the elements of this set. In the example of the power set species one could weight each
subset of $U$ with $k$ elements by $t^k$. The total weight of $P[U]$ would then be
$\sum_{k=0}^n { \binom n k} t^k = (1+t)^n$. The weighted species notion is explained in \cite[Section~2.3]{BLL}.

A multisort species is one for which the label set $U$ is categorized into sorts, and such that
the combinatorial constructions depend on this categorization. In the present exposition the sorts are called colors.
This more advanced notion occurs in \cite[Section~2.4]{BLL}. In the following the species that occur will usually
be weighted multisort species. A species built over a set $U$ with no sort distinction might be called monochromatic.
The case where the focus is on counting the set  $F[U]$  of combinatorial objects is
obtained by setting the weight of each object equal to one. In the power set example the value of $(1+t)^n$ with
$t=1$ is the count $2^n$.

We have seen that in species theory it is useful to have a concept of weighted set. The notation here
is chosen so operations on weighted sets appear very similar to operations on
sets. Fix a commutative ring $R$ (perhaps the real numbers or the complex numbers).
A \emph{weighted set} is a finite set together with a weight function $A$ from the set to
$R$. The finite set is the domain $\dom(A)$, and the function $A$ is from $\dom(A)$ to $R$.

Suppose that $A,B$ are weighted sets. There are natural concepts of disjoint union
$A+B$ and cartesian product $A \times B$.
The disjoint union is produced in the obvious way. If $A: \dom(A) \to R$
is a weighted set, and $B: \dom(B) \to R$ is a weighted set, then
$A + B: \dom(A) + \dom(B) \to  R$ is the weighted set
defined on the disjoint union $\dom(A) + \dom(B)$ of sets by
$(A+B)(\alpha) = A(\alpha)$ for $\alpha$ in $\dom(A)$ and
$(A+B)(\beta) = B(\beta)$ for $\beta$ in $\dom(B)$.
Similarly, the cartesian product $A \times B: \dom(A) \times \dom(B) \to R$
is defined on the cartesian product $\dom(A) \times \dom(B)$ of sets by
the \emph{multiplicative property}
$(A \times B)(\alpha,\beta) = A(\alpha) B(\beta)$.
The case of sets without any special weighting
is obtained by giving each element weight one. In that case
the weighted set may be identified with the set.


If $A: \dom(A) \to R$ is a weighted set, then it has \emph{total weight}
\begin{equation}
|A| = \sum_{\alpha\in \dom(A)} A(\alpha).
\end{equation}
This is an obvious generalization of cardinality. It is obvious
that $|A+B| = |A| + |B|$. When the
multiplicative property holds it is also true that
$|A \times B| = |A| \, |B|$. This follows from the distributive law.
The idea is that the domain of a weighted set is a set of combinatorial
objects. One wants to manipulate these sets of combinatorial objects, while
at the same time keeping track of their weights. Since the weighted
set operations automatically include the corresponding operations on
sets of combinatorial objects, one can treat the weighted sets as sets, but with
extra structure. It may be convenient to think of the elements $\alpha$
of the domain of $A$ as being elements of the weighted set $A$. With this
convention it is possible to write $\alpha \in A$ for an element of the
domain of $A$ and to write $A(\alpha)$ for the weight of this $\alpha$.

In the following it is convenient to consider disjoint sums and cartesian products of indexed
families of weighted sets. Suppose that for each $p$ in some index set there
is a weighted set $A_p$. The weighted set $\sum_p A_p$ has domain set consisting
of all pairs $(p,\alpha)$, where $\alpha$ is in the domain set of $A_p$. The
weight of $(p,\alpha)$ is $A_p(\alpha)$. The weighted set $\prod_p A_p$ has
domain set consisting of all functions $p \mapsto \tau_p$ such that each $\tau_p$ is
in the domain set of $A_p$. The weight of $\tau$ is $\prod_p A_p(\tau_p)$.

A \emph{species} $F$ assigns to each colored set $a: U \to \Xc$ a corresponding set
$\dom(F[a])$ of combinatorial objects together with a weight function $F[a] : \dom(F[a]) \to R$.
In other words, it takes a colored set $a$ to a weighted set $F[a]$.
Furthermore, it is assumed that permutations of the labels that carry along the colors
induce corresponding permutations of the weighted sets that carry along the weights.
The {total weight} of $F[a]$ is obtained by summing the weights of the individual combinatorial
objects in $\dom(F[a])$. The total weight is denoted
\begin{equation}
f(a) = |F(a)| = \sum_{\alpha \in \dom(F[a])}  F[a](\alpha).
\end{equation}
The end product of species theory is the computation of total weights of combinatorial objects
built from colored sets. This is a natural generalization of counting.




For each species $F$ there is a corresponding \emph{exponential generating function} $F(z)$
with tensor coefficients given by the total weights $f(a)$.
The operations on species are designed to give corresponding operations on the exponential generating functions.

Given two species $G$ and $H$, there is another species $G+H$ that
is the \emph{combinatorial sum}. It is defined by $(G+H)[a] = G[a] + H[a]$, where
the sum on the right is weighted set disjoint sum. The corresponding
exponential generating functions satisfy $(G+H)(z) = G(z) + H(z)$.

Given two species $G$ and $H$, there is a species $ G*H$ that is the \emph{combinatorial product}. It is defined by
\begin{equation}
(G * H)[a] = \sum_{ \langle V, W  \rangle } G[a_V] \times H[a_W] .
\end{equation}
This is a disjoint sum of  cartesian products of  weighted sets.
The $\langle V, W \rangle$ that index the disjoint sum range over ordered pairs of subsets with
$V \cap W = \emptyset$ and $V + W = U$. Also $a_V$ denotes
the restriction of $a$ to $V$.
It follows that the exponential generating functions satisfy
\begin{equation}
(G * H)(z) = G(z) H(z).
\end{equation}

Consider two color palettes $\Yc$ and $\Xc$. Say that $G$ is a species on
$\Yc$ colored sets, and $H_y$ for $y$ in $\Yc$ are species
on $\Xc$ colored sets. Suppose that for each $y$ that $H_y(\emptyset) = \emptyset$.
The \emph{combinatorial
composition} $G \circ H$ as a species on $\Xc$ colored sets is defined by
\begin{equation}
(G \circ H)[a] = \sum_\Gamma \sum_{b: \Gamma \to \Yc} G[b] \times \prod_{V \in\Gamma} H_{b(V)}[a_V].
\end{equation}
This is a disjoint sum of cartesian products of weighted sets. The sum is indexed by set partitions
$\Gamma$ of $U$ and colorings $b: \Gamma \to \Yc$ of these partitions. Each cartesian product
is indexed by the blocks of the partition.
The corresponding exponential generating functions satisfy
\begin{equation}
(G \circ H)(z) = G(H(z)).
\end{equation}

The combinatorial composition is complicated enough to make it worth looking at the
construction in more detail. Consider a colored set $a: U \to \Xc$. An element $\alpha$ of the underlying domain set of
the weighted set $(G \circ H)[a]$ is of the form $\alpha = (\Gamma, b, \lambda, \tau)$.
Here the element of the index set for the disjoint sum is given by specifying a set partition $\Gamma$ of $U$ and
a coloring $b: \Gamma \to \Yc$. The $\lambda$ is a structure in the underlying domain set of
$G[b]$. For each block $V$ in $\Gamma$ the value $\tau_V$ is in the underlying domain set of
$H_{b(V)}[a_V]$. The weight of $\alpha$ is the product $G[b](\lambda) \prod_{V \in \Gamma} H_{b(V)}[a_V](\tau_V)$.



An important special case is when $G$ is monochromatic and there is only one $H$ species.
In particular, we may take for $G$ the species $E$ that assigns to each set
a single point with weight one. Then $E \circ H$ is
the combinatorial exponential given by
\begin{equation}
(E \circ H)[a] = \sum_\Gamma \prod_{V \in\Gamma} H[a_V].
\end{equation}
The corresponding exponential generating functions satisfy
\begin{equation}
(E \circ H)(z) = \exp(H(z)).
\end{equation}

There are other useful operations on species. If $F$ is a
species, then there is a species $F'_x$ given by
\emph{combinatorial derivative}. It is defined by
\begin{equation}
F'_x[a] = F[a^x],
\end{equation}
where $a^x: U + \{*\} \to \Xc$ is equal to $a$ on $U$ and
has value $x$ on $*$. Here $*$ is a label point that is not in $U$.
Then the exponential generating function satisfies
\begin{equation}
F'_x(z) = \frac{\partial F(z)}{\partial z_x}.
\end{equation}
The combinatorial derivative is perhaps best regarded as a partial derivative,
where the coordinate index is the color $x$. In the following the index may
be used either as a subscript or as a superscript, depending on the context.
This represents an attempt to follow tensor calculus conventions.


\subsection{Feynman diagram species}

A first notion of Feynman diagram species is the following.
Fix a color palette $\Xc$. For each colored set $a: U \to \Xc$
there is a set $Z[U]$ of  vacuum Feynman diagrams with label set $U$. Each
diagram $\gamma$ consists of a set partition $\Gamma$ of $U$ and a perfect matching $\sigma$ of $U$.
The weight of a diagram is $\Feyn(a,\gamma)= S(a,\Gamma) C(a,\sigma) $. The
weighted set of all such diagrams is $Z[a]$.


A vacuum Feynman diagram is \emph{connected} if the
label set cannot be partitioned into blocks that are not coupled by the diagram.
Let $Z$ be the species of vacuum Feynman diagrams, and let $F$ be the species of connected
vacuum Feynman diagrams. By the general theory,
\begin{equation}
Z = E \circ F.
\end{equation}
This says that every vacuum
Feynman diagram gives rise to a set partition on each block of which there is a
connected vacuum Feynman diagram. The exponential generating function
identity is $Z(z) = \exp(F(z))$.

The simplest connected vacuum Feynman diagrams are for $n=2$. There is a diagram with one vertex and a single loop;
there is also a diagram with two vertices and a line that forms a bridge between them. The next simplest case
is with $n=4$. There is a diagram with one vertex and a double loop; there is a diagram with two vertices and
two lines joining them; there is a diagram with two vertices, one loop, and a line that forms a bridge between
the vertices; there is a diagram with three vertices and two bridge lines.

The weight of each diagram is given by the Feynman rules. For instance, consider the $n=4$ vacuum
diagram with two vertices and two lines joining them. Say more specifically that the underlying
set is $U = \{i,j,k,\ell\}$ and the partition has blocks $\{i,j\}$ and $\{k,\ell\}$.
The lines are $\{i,k\}$ and $\{j,\ell\}$. The weight is then $S_{a_i\, a_j} C^{a_i\, a_k} C^{a_j\, a_\ell} S_{a_k\, a_l}$
For another example, consider the $n=4$ diagram with two vertices, one loop, and a bridge. Say that
the partition has blocks $\{i\}$ and $\{j, k, \ell\}$ and the lines are the bridge $\{i,j\}$ and the
loop $\{k, \ell\}$. The weight for this diagram is $S_{a_i} C^{a_i\, a_j} S_{a_j\, a_k\, a_\ell} C^{a_k \,a_\ell}$.

\subsection{Indicator species}

Indicator species are combinatorial constructions that return at most one
object. They are not particularly interesting in themselves, but they are
excellent building blocks for more complicated species.
The simplest example is the monochromatic set indicator species $E$. Given a label set $U$, the
value of $E[U]$ is a single point of weight one. The corresponding exponential
generating function is a function of a single variable $w$; it is just the
exponential function $E(w) = \exp(w)$.

There are also indicator species that act on colored sets. We shall need the
one point of color $x$ indicator species
$X_x$  that  returns a single point of weight one, but only provided that
the input is a colored set with one element of color $x$. For every other input
the return is the empty set. The exponential generating function of $X_x$ is
the coordinate function $X_x(z) = z_x$.

A  more interesting example is the \emph{line indicator species} $L$. For a colored label set
$a: \{i,j\} \to \Xc$ with two points it gives a single point with weight $C^{a_i a_j}$.
Otherwise it returns the empty set.
The line indicator species has quadratic exponential generating function $L(z) = \frac{1}{2} \sum_x \sum_y C^{xy} z_x z_y$.
The perfect matching species $\Mat$ satisfies
\begin{equation}
\Mat = E \circ L.
\end{equation}
This already shows the power of the indicator species concept.

It will be useful to have the combinatorial partial derivative $L'^x$ of the line indicator species.
The value of this on a one-point colored set $a: \{j\} \to \Xc$ is
a single point  with weight $C^{x a_j}$. In all other cases it returns the empty set.
This is a species that recognizes a single point, but regards it as
having a line from an external point of color $x$. The exponential generating
function of this species is $L'^x(z) = \sum_y C^{xy} z_y$.


\subsection{Connected Feynman diagrams with a leg or amputated line}

Here are two useful species. The first is the species $F^{\uparrow x}$ of connected Feynman diagrams
with an external leg of color $x$. Given a colored set $a: U \to \Xc$, this species consists of all
connected Feynman diagrams $\tau$ on the disjoint union $U + \{*\}$, where $*$ is an external point that is a leg label of color $x$.
Each connected diagram $\tau$ has a set partition $\Gamma$ of $U$ and a perfect matching $\sigma$ of $U + \{*\}$.
The weight of a diagram is $\Feyn(a^x, \tau) = S(a^x, \Gamma) C(a^x, \sigma)$. These weighted diagrams define
the weighted set $F^{\uparrow x}[a]$. The total weight of this weighted set contributes to the coefficient of
the $z^a$ term in the exponential generating function. The external point $*$ plays a role similar to that
of the external point in the definition of the derivative species, except that only the perfect matching
part of the diagram extends to this point.


The second is the species of $F^\odot_y$ of connected Feynman diagrams with a amputated line
of color $y$. This assigns to each colored set $a: U \to \Xc$ a set partition $\Gamma$ of $U$,
an amputated line label $j$ in $U$ with $a_j = y$, and a perfect matching $\sigma_j$ of the difference set $U - \{j\}$,
and this is done in such a way as to produce a connected diagram $\gamma_j$.
The weight of a diagram is $\Feyn(a, \gamma_j) = S(a, \Gamma) C(a, \sigma_j)$. These weighted diagrams define
the weighted set $F^{\odot}_y[a]$. Again the total weight of this weighted set contributes to the coefficient of
the $z^a$ term in the exponential generating function. This construction is somewhat like the combinatorial
product construction in the special case when the disjoint union $U = \{j\} + (U - \{j\})$ involves
one-point subsets. Again only the perfect matching part of the diagram is restricted.


\begin{lemma} There is a species isomorphism
\begin{equation}
F^{\uparrow x}  = L'^x \circ F^\odot.
\end{equation}
\end{lemma}

The proof is to note that the weighted set $(L'^x \circ F^\odot)[a]$ is computed with a trivial partition with
only one block $U$. An object in the underlying set is given by $U$, a color $b(U)$, and
a connected Feynman diagram $\gamma_j$ with some specified amputated line label $j$ in $U$ with $a_j = b(U)$. The weight
is $C^{x b(U)} \Feyn(a,\gamma_j) =C^{x a_j} \Feyn(a,\gamma_j) $.  Map this to a connected diagram $\tau$ by adding a line from $j$ to
an external leg label $*$ of color $x$. This map may be reversed by defining $j$ to be the vertex
adjacent to $*$. The weight $\Feyn(a^x,\tau)$ is the same.
This gives an object in the weighted set  $F^{\uparrow x}[a]$.


The lemma says that a connected Feynman diagram with a leg to an external point is obtained by
adjoining a line to a connected Feynman diagram with an amputated line.  The exponential
generating function identity is $F^{\uparrow x}(z) = \sum_y C^{xy} F^\odot_y(z)$.

It is easy to generate examples of connected diagrams with single amputated lines. For $n=1$ there is
a single vertex with an amputated line. For  $n=3$ there is the one vertex diagram with loop and amputated
line; also there is the two vertex line with bridge line and one amputated line. For $n=5$
there is a diagram with one vertex and a double loop and one amputated line, and there is a diagram with two vertices and
two lines joining them and one amputated line. However now there are two kinds of diagrams with two vertices, one loop, and a line that forms a bridge between the vertices, since one can place the amputated line in the vertex with the loop or the vertex
without the loop. Also, there are two kinds of diagrams with three vertices and two bridge lines, since one can place the
amputated line in the center vertex or in an end vertex.

Adding a line and an external point
to an amputated diagram gives the corresponding diagram with a leg line to an external point.
The simplest such diagram has a one point vertex with a line to an external point.
The next simplest such diagram has a loop and a line to an external point; in the
physics literature this is sometimes called a ``tadpole'' diagram.


\section{Line-irreducible diagrams}

\subsection{Lumps and bridges}

A line in a connected Feynman diagram is a \emph{bridge} if removing it
produces a diagram that is not connected. When all bridges are removed
the resulting diagram will  have amputated lines. The label set
may then be partitioned into connected components on each one of which
is a connected diagram with amputated lines.

A connected diagram with amputated lines is a \emph{line-irreducible diagram} if
it has no bridge.  This notion is closely related to the graph theory notion of 2-edge-connected graph. It is
also a possible formulation of the physics concept of 1-particle-irreducible diagram.

Some obvious examples of line-irreducible diagrams are the line-irreducible vacuum diagrams.
For instance, there is the $n=2$ vertex diagram with one vertex and a loop. Also, there is the $n=4$
vacuum diagram with one vertex and a double loop, as well as the $n=4$  vacuum diagram
with two vertices and two lines connecting them. There are also non-vacuum diagrams with
a single truncated line. The simplest one is the $n=1$ diagram with one vertex and a truncated line.
There are also $n=3$ and $n=5$ diagrams obtained by adding a truncated line to the above vacuum diagrams.
One can go on and construct diagrams with two truncated lines; the same procedure gives examples
with $n=2,4,6$.


Every connected vacuum Feynman diagram may be decomposed as a tree, where the vertices of the tree are line-irreducible
diagrams with amputated lines, and the edges of the tree are the bridges. Each constituent line-irreducible
diagram is called a \emph{lump}. A lump can consist of a single point.

Write $M^\odot$ for the species of line-irreducible diagrams. The line-irreducible diagrams
on label set $U$ consist of a set partition of $U$ (the vertices) and a perfect matching of
a subset of $U$ (the lines) with the condition that there is no bridge.

The species $M^\odot$ decomposes into two
parts $M^\odot = M^\odot_0 + M^\odot_1$.  The part $M^\odot_0$ gives line-irreducible diagrams with no
one-point vertices. If a line-irreducible diagram has one-point vertices, then it consists
of a single one-point vertex. Thus the other part $M^\odot_1$ consists of line-irreducible diagrams consisting of
a single label point belonging to a one-point vertex.

There is another species  $M$ related to the physics concept of 1-particle-irreducible
diagram. The diagrams are obtained from line-irreducible Feynman
diagrams by replacing each amputated line label with an ordinary label together with
a line to a leg label. In the following construction these legs are used to construct
bridges. For that reason, the species $M$ will be called the species of \emph{line-irreducible
Feynman diagrams with bridge legs}. A leg label that is adjoined by a line to the original line-irreducible
diagram is called a \emph{bridge leg label}. The label set $U = U_0 + U_\beta$, where the labels in $U_0$ belong to vertices, while each bridge leg label in $U_\beta$ is attached to a label in $U_0$ by a line.

The species $M = M_0 + M_1$ also decomposes into two parts. The part $M_0$ gives diagrams with
no one-point vertices. The part $M_1$ consists of diagrams with a line from a label belonging
to a one-point vertex to a bridge leg label.




\subsection{Bridge identities}

The context for the following results is the species $F$ of connected Feynman diagrams
and variations on this\break species. One variation is the species of connected Feynman diagrams
with a designated bridge. This produces ordered pairs consisting of a  Feynman diagram
and a bridge of the diagram.
Another variation is the species of connected  Feynman diagram with a designated lump. This produces
ordered pairs consisting of a Feynman diagram and a lump of the diagram.
These  species are less symmetrical than the species $F$;  in compensation they enter in recursive structures.

\begin{lemma}
Let $F^\leftrightarrow$ be the species of connected Feynman diagrams with a designated bridge.
Let $L$ be the line indicator species. For each $y$ let $F^\odot_y$ be the species of connected diagrams with
an amputated line of color $y$. Then
\begin{equation}
F^\leftrightarrow = L \circ F^\odot .
\end{equation}
\end{lemma}

To prove this lemma, start with $(L \circ F^\odot)[a] $. The elements of this weighted set
are indexed by two-element partitions $\Delta = \{V,W\}$ of $U$ together with colorings
$b: \Delta \to \Xc$. Consider a partition $\Delta$. A typical element is a diagram $\gamma$ on $V$ with some amputated line label
$j$ of color $a_j = b(V)$ and a diagram $\delta$ on $W$ with some amputated line label $k$ of
color $a_k = b(W)$. The weight is $C^{b(V)b(W)} \Feyn(a_V,\gamma) \Feyn(a_W, \delta)$. The values
$b(V)$ and $b(W)$ are arbitrary, so one may think of a typical element as a diagram $\gamma$ on $V$
with some amputated line label
$j$  and a diagram $\delta$ on $W$ with some amputated line label $k$. The weight is $C^{a_j a_k} \Feyn(a_V,\gamma) \Feyn(a_W, \delta)$.
Map this pair of diagrams $\gamma, \delta$ to a combined diagram $\sigma$ on $U$ with an additional line from $j$ to $k$.
Its weight $\Feyn(a,\sigma)$ is the same. The designated bridge is the unique line that connected the partition.
The conclusion is that one gets the weighted set $F^\leftrightarrow[a]$.
The map is reversible; given $\sigma$ and a bridge, there is a partition $\Delta = \{V,W\}$ determined by
the connected diagrams $\gamma,\delta$ on each side of the bridge.
The exponential generating function identity is
$F^\leftrightarrow(z) = \frac{1}{2} \sum_x \sum_y F^{\odot}_x(z) C^{xy}  F^\odot_y(z) $.

\begin{lemma}
Let $F^{\bigcirc\leftrightarrow}$ be the species of connected vacuum Feynman diagrams with
a designated incident lump-bridge pair. Then
\begin{equation}
F^{\bigcirc\leftrightarrow} =   F^\odot * F^\uparrow = F^\odot * (L' \circ F^\odot) .
\end{equation}
\end{lemma}


In the statement of the lemma the right hand side should be
interpreted as $\sum_x  F^{\odot }_x * F^{\uparrow x} = \sum_x  F^{\odot }_x * (L'^{x} \circ F^\odot )$.
On a colored set $a: U \to \Xc$.
this gives an ordered pair of subsets $V,W$
that are disjoint and exhaust $U$. Furthermore, a typical object is given by a connected diagram $\gamma$ on $V$
with amputated line label $i$ and a connected diagram $\delta$ on $W$ with external leg label. This
external leg label may be identified with $i$.  The only $x$ that contributes is $x = a_i$.
The weight of this pair of diagrams is $\Feyn(a_V, \gamma) \Feyn(a_W,\delta)$. The two diagrams
are mapped to a combined diagram $\sigma$ on all of $U$. The weight of this diagram is $\Feyn(a,\sigma)$, which
is the same. The $V$ subset is the one that has the lump as part of it; the bridge connects
$V$ with $W$. The mapping may be reversed, given $\sigma$ and the lump-bridge pair, let the connected diagram on the
lump side of the bridge define $V$ and let $W$ be the complement.
The exponential generating function identity is
$F^{\bigcirc\leftrightarrow}(z) =
 \sum_x  F^{\odot }_x(z) F^{\uparrow x}(z)= \sum_x \sum_y F^{\odot}_x(z) C^{xy}  F^\odot_y(z) $.



\subsection{Lump identities}


Let $M$ be the species of line-irreducible Feynman diagrams
with bridge legs. Regard $M$ as a species with color palette
$\Xc \times \{0, \beta\}$ with another dimension. The 0 and $\beta$ are used to denote
ordinary labels and bridge leg labels.
A coloring is a pair $a: U \to \Xc$ and $c: U \to \{0, \beta\}$.
Specifying $c$ is the same as specifying a disjoint union decomposition $U = U_0 +  U_\beta$.
Then $M[a,c]$ consists of all line-irreducible Feynman diagrams with bridge legs
such that the ordinary labels are in $U_0$ and the bridge leg labels are in $U_\beta$.
Since there are two kinds of labels, there are two kinds of variables. Write them as $z_{x0} = z_x$ and
$z_{x\beta}= K_x$. The exponential generating function $M(z,K)$ depends on both kinds of variables.

As before, let $F^{\odot}_y$ denote the species of connected Feynman diagrams with an amputated
line of color $y$. Define a family of species $ F^*$ indexed by $\Xc \times \{0,\beta\}$ by
$F^{*}_{ y \beta} = F^{\odot}_y$ and $ F^{*}_{ y 0} = X_{y}$.


\begin{lemma}
Let $F^\bigcirc$ be the species of connected vacuum Feynman diagrams with a designated lump.
Let $F^*$ be the two-sort family that consists either of the species of connected Feynman diagrams with one
amputated line or of the one-point indicator species. Then
\begin{equation}
F^\bigcirc = M \circ F^*.
\end{equation}
\end{lemma}

The idea of the proof is that a connected vacuum Feynman diagram together with
a specified lump gives a set partition of the underlying set. The lump defines a
collection of bridges to connected diagrams supported on a corresponding collection of disjoint subsets.
The partition consists of one-point blocks that correspond to the labels in the lump and of other blocks
that correspond to the subsets that support connected diagrams.

The proof begins by fixing a colored set $a: U \to \Xc$.
Consider $(M \circ F^*)[a]$. Each partition $\Delta$ of $U$ and each coloring
$b: \Delta \to \Xc$ and $c: \Delta \to \{0,\beta\}$ gives a product space
$M[b,c] \times \prod_{V \in \Delta} F^{*}_{b(V)c(V)}[ a_V]$. This product space
may be written as $M[b,c] \times \prod_{V \in \Delta_0} X_{b(V)}[a_V] \times \prod_{V \in \Delta_\beta} F^{\odot}_{b(V)}[a_V]$.
This product space is empty unless the blocks $V \in \Delta_0$ are one point sets $V = \{i\}$
with $b(V) = a_i$. It is also empty unless the blocks $V \in \Delta_\beta$  each give a collection
of connected diagrams with corresponding amputated line labels $j$ in $V$ of color $a_j = b(V)$.
Identify the blocks $V=\{i\} \in \Delta_0$ with the corresponding $i$, and identify the blocks
$V \in \Delta_\beta$ with the corresponding $j$.
An element of $M[b,c]$, originally a line-irreducible diagram on $\Delta_0$ with bridge leg labels $V \in \Delta_\beta$,
is identified with a line-irreducible diagram $\gamma$ with ordinary labels consisting of
the points $i$ with $\{i\}$ in $\Delta_0$ and with bridge leg labels $j \in V \in \Delta_\beta$.
An element of the  product space also has a diagram $\delta_V$ for each
$V$ in $\Delta_\beta$. Putting all these diagrams together gives a connected vacuum Feynman
diagram $\tau$ on $U$ with the correct weight. The colored partition also singles out a lump of this diagram. The result is
that these objects build the weighted set $F^\bigcirc[a]$. On the level of exponential generating functions the lemma says that $F^\bigcirc(z) = M(z, F^\odot(z))$.



\subsection{The dissymmetry theorem for connected Feynman diagrams}

The next topic is symmetry about
the center of a tree. A tree is a connected graph without cycles.
A leaf of a tree is a vertex of degree zero. The center of a tree is defined recursively by pruning leaves.
Remove each leaf along with its incident edges to produce a smaller tree.
This process terminates when all the vertices are leaves. There remains a center
vertex or two vertices with a center edge. There is always a center element, either
a vertex or an edge.

There are combinatorial objects that violate this symmetry.
Specifying a distinguished vertex or a distinguished edge in a tree
introduces a lack of symmetry, since this distinguished object may not be at
the center. The following results are based on a general dissymmetry
theorem for trees \cite{BLL}. This theorem relates the tree to these other less
symmetric objects.

Let $T$ be the species of trees; for each set $U$ there is an associated
set $T[U]$ of trees with vertex set $U$. Let $T^\bullet$ be the species of trees with
distinguished vertex; in this case for each $U$ there is set $T^\bullet[U]$
consisting of  ordered pairs: a point in $U$ and a tree on $U$.
Similarly, let $T^-$ be the species of trees with a distinguished
edge. Finally, let $T^{\bullet -}$ be the species of trees with a distinguished
incident vertex-edge pair. The dissymmetry theorem for trees states that
$T + T^{\bullet -} = T^\bullet + T^{-}$.

Here is the proof. For the moment refer to an edge or a vertex of the tree as an element. Let
$T^\star$ be the species of trees with a distinguished element. It is clear that
$T^\star = T^\bullet + T^-$.
Let $T^{\star \neq c}$ be the species of trees with a distinguished element other than the center.
Then $T^{\star \neq c} = T^{\bullet -}$. The map between these species maps each element
other than the center to the pair consisting of the element and the incident element that
is closer to the center. To reverse the correspondence, map each
pair of incident elements to the element in the pair farthest from the center.
On the other hand, $T + T^{\star \neq c} = T^\star$, since specifying a tree automatically
specifies its center. These equations combine to give the desired result
$T + T^{\bullet -} = T + T^{\star \neq c} = T^\star = T^\bullet + T^{-}$.



The dissymmetry theorem for connected Feynman diagrams is a variation on
the dissymmetry theorem for trees. In this case the vertices of the
trees are the lumps, and the edges of the tree are the bridges.  Since a
connected Feynman diagram determines its center lump or edge, this gives
a relation for the species of connected Feynman diagrams.

\begin{proposition}
Let $F$ be the species of connected vacuum Feynman diagrams. There is a species isomorphism
\begin{equation}
F + F^{\bigcirc\leftrightarrow}   =  F^\bigcirc + F^\leftrightarrow.
\end{equation}
\end{proposition}

In a previous section it was shown that these species with designated
structures were equivalent to certain species defined as compositions.
This leads to the following statement of the dissymmetry theorem.

\begin{theorem}[\sc Dissymmetry theorem] The species $F$ of connected vacuum Feynman
diagrams satisfies the identity
\begin{equation}
F +    F^{\odot} * (L' \circ F^\odot)  = M \circ  F^*  + L \circ F^\odot .
\end{equation}
\end{theorem}

In terms of exponential generating functions this says that
\begin{multline}
F(z) + \sum_x \sum_y  F^{\odot}_x(z) C^{xy}  F^{\odot}_y(z) =  M(z,  F^\odot(z)) \\+ \frac{1}{2}\sum_x \sum_y  F^{\odot}_x(z) C^{xy}F^{\odot}_y(z) .
\end{multline}


The dissymmetry theorem may be expressed in a more detailed form. Recall that
$M = M_0 + M_1$, where $M_1$ has a special form. Namely, $M_1[a,c]$ is empty
unless $U = \{i,j\}$ with $c_i = 0$ and $c_j = \beta$. In that case it consists of diagrams
 with a one-point vertex at $i$ and a bridge leg label at $j$. The
weight of this diagram is $S_{a_i} C^{a_i a_j}$. The exponential generating function
of the species $M_1$ is $M_1(z,K) = \sum_x \sum_y z_x S_x C^{xy} K_y$.



\begin{corollary}
The species of connected vacuum Feynman diagrams satisfies
\begin{equation}
F +   F^{\odot} * (L' \circ F^\odot)  = M_0 \circ  F^* + M_1 \circ F^* + L \circ F^\odot .
\end{equation}
\end{corollary}

\begin{proposition}
The composition $M_1 \circ F^*$ acting on a colored set $a: U \to \Xc$ gives
all connected diagrams for which there is a distinguished point $i$
that belongs to a one-point vertex.
\end{proposition}


The corollary thus leads to the exponential generating function identity
\begin{multline}
F(z) + \sum_x \sum_y  F^{\odot}_x(z) C^{xy}  F^{\odot}_y(z) =  M_0(z,  F^\odot(z)) \\+
\sum_x z_x S_x  C^{xy} F^\odot_y(z) + \frac{1}{2}\sum_x \sum_y  F^{\odot}_x(z) C^{xy}F^{\odot}_y(z) .
\end{multline}


\subsection{The fixed point equation for connected Feynman dia-\break grams}

Recall that $F^{\uparrow x}$ is the species of connected Feynman diagrams with
an external leg label of color $x$. Similarly, $M^{\uparrow x}$ is defined as
the species of line-irreducible diagrams with bridge legs and also with
a leg to an external leg label of color $x$.  Let $a: U \to \Xc$ and $c: U \to \{0,\beta\}$.
A diagram $\gamma$ contributing to $M^{\uparrow x}[a,c]$
consists of a partition $\Gamma$ of the set $U_0$ where $c = 0$ and a perfect matching $\sigma$
of $U + \{*\}$. The bridge points in the set $U_\beta$ where $c = \beta$ are matched
to points in $U_0$, and the extra point $*$ is also matched with a point in $U_0$. The
amputated diagram with the bridge points and the extra point removed is required to be
line irreducible. The weight of the diagram is $\Feyn(a^x,\gamma)$. The following
theorem is based on the observation that a connected diagram with a leg to
an external point singles out a lump with a leg to an external point and with
bridge legs each of which is attached to a connected diagram with amputated line.


\begin{theorem}[\sc Fixed point equation] The species of connected Feynman diagrams with an external leg
satisfies the fixed point equation
\begin{equation}
F^{\uparrow x} =  L'^x \circ F^\odot =  M^{\uparrow x} \circ  F^{*}.
\end{equation}
\end{theorem}

Here is a proof sketch. To analyze the composition, begin as usual with $a: U \to \Xc$.
The weighted set  $(M^{\uparrow x} \circ F^*)[a]$ is then a disjoint sum over set partitions
$\Delta$ of $U$ and colorings $b: \Delta \to \Xc$ and $c: \Delta \to \{0,\beta\}$
of products
$M^{\uparrow x}[b,c] \times \prod_{V \in \Delta_0} X_{b(V)}[a_V] \times \prod_{V \in \Delta_\beta} F^{\odot}_{b(V)}[a_V]$.
Identify each block $V=\{i\} \in \Delta_0$ with the corresponding $i$, and identify each block
$V \in \Delta_\beta$ with the corresponding amputated line label $j$ in $V$.
An element of $M^{\uparrow x}[b,c]$
is identified with a line-irreducible diagram $\gamma$ with ordinary labels consisting of
the points $i$ with $\{i\}$ in $\Delta_0$, with bridge leg labels $j \in V \in \Delta_\beta$, and
with one more leg label $*$. There is a line from some $k$ to $*$.
An element of the  product space also has a diagram $\delta_V$ for each
$V$ in $\Delta_\beta$. Putting all these diagrams together gives a connected vacuum Feynman
diagram $\tau$ on $U + \{*\}$ with weight that includes a factor $C^{x a_{k}}$.
Such diagrams build the weighted set $F^{\uparrow x}[a]$.
In terms of exponential generating functions the theorem says that
\begin{equation}
F^{\uparrow x}(z) = \sum_x C^{xy} F^{\odot}_y(z)  = M^{\uparrow x}(z, F^{\odot}(z) ).
\end{equation}

The decomposition $M = M_0 + M_1$ gives a corresponding decomposition $M^{\uparrow x} = M^{\uparrow x}_0 + M^{\uparrow x}_1$.
Here $M_0^{\uparrow x}$ is the species of line-irreducible diagrams with bridge legs that include
a leg to an external point of color $x$ and that have no one-point vertices.
The other species $M_1^{\uparrow x}$ is the species that only gives a single line
from a  point $i$ belonging to a one-point vertex to an external point $*$
 of color $x$.
  Namely, $M^{\uparrow x}_1[a,c]$ is empty
unless $U = \{i\}$ with $c_i = 0$. In that case it consists of a
line with a one-point vertex at $i$ and an external leg label at $*$ of color $x$. The
weight of this diagram is $S_{a_i}  C^{a_i x}$. There are no bridge leg labels, and so
composition with this species is trivial.

\begin{proposition}
The composition $M_1^{\uparrow x} \circ F^*$ acting on a colored set $a: U \to \Xc$ returns
the empty set, unless $U = \{i\}$ is a one-point vertex. It that case it give
a diagram on $U + \{*\}$  with a line from $i$ to *. The weight
of the diagram is $S_{a_i}  C^{a_i x}$.
\end{proposition}

With a slight abuse of notation one may denote $M_1^{\uparrow x} \circ F^*$ by
$M_1^{\uparrow x}$. The exponential generating function is just $M_1^{\uparrow x}(z) = \sum_y C^{xy} S_y z_y $.

\begin{corollary} The species of connected Feynman diagrams with an external leg satisfies the fixed
point equation
\begin{equation}
F^{\uparrow x} =  L'^x \circ F^\odot =  M_0^{\uparrow x} \circ  F^{*} + M_1^{\uparrow x}.
\end{equation}
\end{corollary}

In terms of exponential generating functions this says that
\begin{equation}
F^{\uparrow x}(z) = \sum_x C^{xy} F^{\odot}_y(z)  = M_0^{\uparrow x}(z, F^{\odot}(z) ) + \sum_y C^{xy} S_y z_y .
\end{equation}


\section{The Legendre transform}


\subsection{The Legendre transform for connected Feynman dia-\break grams}

The results above give  combinatorial identities
for connected vacuum Feynman diagrams. In order to make the connection
with the Legendre transform, it is useful to replace one-point vertices
by leg labels, so there is no longer a factor $S_x$ attached to such
a vertex. Furthermore, the leg labels are specified as part of the
color structure. The resulting species provides a more detailed
combinatorial description. Henceforth $F$ denotes this species of connected Feynman diagrams with legs.

Given $a: U \to \Xc$ and $c: U \to \{0,\lambda\}$, the
corresponding weighted set $F[a,c]$ consists of all connected Feynman diagrams
with ordinary labels on the subset $U_0$ where $c = 0$ and with leg labels
on the subset $U_\lambda$ where $c = \lambda$. A diagram $\gamma$ consists
of a set partition $\Gamma$ of $U_0$ into blocks of size two or more and a perfect matching $\sigma$ of $U$,
with the constraint that these structures connect the label set $U$.
The weight of such a diagram is $\Feyn(a,\gamma) = S(a,\Gamma) C(a,\sigma)$.
The reason this works is that a partition of $U$ is equivalent to
a subset of $U$ together with a partition of the complement into blocks of size
two or more. The sum over subsets arising from the various colorings $c$ together with the sum over the partitions of
the complement into blocks of size two or more give the sum over
all partitions.

Denote the variables for the colors by $z_{x0} = z_x$ and by $z_{x \lambda} = J_x$.
The $J_x$ variables that occur in the exponential generating function
compensate the missing $S_x$ parameters in the action, so the
effect is to replace every $S_x z_x$ by $J_x$.
The derivative species of $F$ with respect to a leg label of color $x$
will be denoted $F'^x$. This replaces the species $F^{\uparrow x}$ considered
before. The value of $F'^x$ on a colored set is a connected
Feynman diagram on the set $U$ augmented by an external point $*$ of color $x$ that
is a leg label. That is, $F'^x[a,c] = F[a^x,c^\lambda]$, where $a^x$ takes
value $x$  and $c^\lambda$ takes value $\lambda$
on $*$.  It follows that the exponential generating functions
are $F(z,J)$ and $F'^x(z,J) = \partial F(z,J)/\partial J_x$.

The species $F^\odot_y$ of connected Feynman diagrams with a single amputated leg
label of color $y$ is defined similarly. Give $a$ and $c$ the corresponding diagrams
are given by a choice of $j$ in $U$ with $a_j = y$, a partition $\Gamma$ of $U_0$, and a perfect matching
of $U - \{j\}$.


In the dissymmetry theorem there are now two kinds of leg labels, and in the appropriate
places it is natural to employ a color palette $\Xc \times \{0,\lambda,\beta\}$. The leg labels
corresponding to $\lambda$ come from the one-point vertices, while the leg labels corresponding
to $\beta$ are the bridge leg labels that come from the decomposition into line-irreducible diagrams.
The relevant species is $M$, the species of line-irreducible diagrams with bridge legs. This is
defined for a colored set with colorings $a: U \to \Xc$ and
$c: U \to \{0,\lambda,\beta\}$. The coloring $c$ defines a   decomposition
$U = U_0 + U_\lambda + U_\beta$. The resulting diagrams each consist of a partition $\Gamma$ of
$U_0$ and a perfect matching of $U$. Each bridge label in $U_\beta$ is connected by a line
to $U_0 + U_\lambda$. Each diagram is line-irreducible on $U_0 + U_\lambda$.

The remarkable fact is that $M = M_0 + M_1$, where the two component species have entirely different
characters. The species $M_0$ gives line-irreducible diagrams with bridge legs
for which $U_\lambda$ is empty, while the species
$M_1$ gives line-irreducible diagrams with bridge legs for which $U_\lambda$ is non-empty. However
the only $M_1$ diagrams are those that involve a line from a $\lambda$ leg label to a $\beta$ bridge
label. The weight of such a diagram with $c_i = \lambda$, $c_j = \beta$ is $C^{a_i\,a_j}$.


Set $K_y = F^\odot_y$.  Then $K^*$ is a family of species such that
$K^*_{y0} = X_{y0}$, $K^*_{y\lambda} = X_{y\lambda}$, and $K^*_{y \beta} = K_y$. The following
result is a restatement of the previous dissymmetry theorem, except that Feynman diagrams
are on a label set including $\lambda$ leg labels.


\begin{theorem}[\sc Dissymmetry theorem]
The species $F$ of connected Feynman diagrams with legs satisfies
\begin{equation}
F +   K * (L'\circ K)= M_0 \circ  K^* + M_1 \circ K^* + L \circ K ,
\end{equation}
where $K = F^\odot$.
\end{theorem}

In terms of exponential generating functions this says that
\begin{multline}
F(z,J) + \sum_x \sum_y  K_x C^{xy}  K_y =  M_0(z, K) \\+
\sum_x J_x C^{xy} K_y + \frac{1}{2}\sum_x \sum_y  K_x C^{xy}K_y ,
\end{multline}
where $K_y = F^\odot_y(z,J)$. This  equation exhibits the duality that
underlies the Legendre transform. The $J_x$ parameters play the role of
the one-point vertices. The $K_y$ represent the
contribution of connected graphs with an amputated line. The $JCK$
term in the formula is the dual pairing of these quantities. The
duality is mediated by the line corresponding to covariance $C^{xy}$.


If $F$ is the species of connected Feynman diagrams with legs, then $F'^x$ is the
species of connected Feynman diagrams with legs and with an additional line to an external leg label
of color $x$. Similarly, $F^\odot_y$ is the species of connected Feynman diagrams with
legs and with one amputated line label of color $y$. The species $M$ of line-irreducible diagrams
with legs and bridge legs has a derivative species $M'^{x}$ whose value on $a,c$ is the value of
$M$ on the $a^x, c^\beta$ that color the extra point * with colors $x,\beta$. This in turn has
a decomposition $M'^{x} = M'^{x}_0 + M'^x_1$. The
species $M'^{x}_0$ is the species of line-irreducible diagrams with bridge legs (but no other legs)
that include a line to an external bridge leg label of color $x$. Similarly, $M'^x_1$ is the species of lines from a leg label to an
external bridge leg label of color $x$. The weight of such a single line diagram with $c_i = \lambda$ is
$C^{a_i\,x}$. The relation between these species is given by the following result.

\begin{theorem}
The species of connected Feynman diagrams with legs satisfies the fixed point equation
\begin{equation}
F'^{x} = L'^x \circ K =    M'^{x}_0 \circ  K^* + M'^{x}_1,
\end{equation}
where $K = F^{\odot}$.
\end{theorem}

In terms of exponential generating functions this says that
\begin{equation}
\frac{\partial F(z,J)}{\partial J_x} = \sum_x C^{xy} K_y(z,J)  = \frac{\partial M_0(z, K(z,J) )}{\partial K_x} + \sum_y C^{xy} J_y,
\end{equation}
where $K_y = F^\odot_y(z,J)$.







\subsection{The Legendre transform for exponential generating functions}

The formulas in the physics literature are obtained from the exponential
generating function identities by setting each $z_x=1$ and then
suppressing these variables from the notation. The dissymmetry theorem then takes
the form
\begin{equation}
F(J)=   M_0(K) + JCK - \frac{1}{2} KCK .
\end{equation}
The fixed point theorem is
\begin{equation}
\frac{\partial F(J) }{\partial J_x} = \sum_y C^{xy} K_y = \frac{\partial M_0(K)}{\partial K_x} + \sum_y C^{xy} J_y.
\end{equation}

Define
\begin{equation}
W(K) = \frac{1}{2} KCK - M_0(K).
\end{equation}

The Legendre transformation in terms of covariant variables is
\begin{equation}
F(J) + W(K) =  JCK.
\end{equation}
where
\begin{equation}
\frac{\partial F(J)}{\partial J_x} = \sum_y C^{xy} K_y.
\end{equation}
and
\begin{equation}
\frac{\partial W(K)}{\partial K_x} = \sum_y C^{xy} J_y.
\end{equation}
To go from $F(J)$ to $W(K)$ one needs to solve for $J$ in terms of $K$ by
inverting the first equation. To go from   $W(K)$ to $F(J)$ one needs to solve for $K$ in terms of $J$ by
inverting the second equation.


It is common to use variables $\phi = CK$. Define the effective potential
$S(\phi) = W(K)$. If we write $I(\phi) = M_0(K)$, then this gives
\begin{equation}
S(\phi) =  \frac{1}{2} \phi C^{-1} \phi - I(\phi).
\end{equation}
One possible reason for hesitating to use these variables is to avoid the awkward inverse covariance $C^{-1}$.
The Legendre transformation in terms of these variables is
\begin{equation}
F(J) + S(\phi) = J\phi.
\end{equation}
where
\begin{equation}
\frac{\partial F(J)}{\partial J_x} = \phi^x.
\end{equation}
and
\begin{equation}
\frac{\partial S(\phi)}{\partial \phi^x}= J_x.
\end{equation}
To go from $F(J)$ to $S(\phi)$ one needs to solve for $J$ in terms of $\phi$ by
inverting the first equation. To go from   $S(\phi)$ to $F(J)$ one needs to solve for $\phi$ in terms of $J$ by
inverting the second equation.

\section{Appendix: Abstract Feynman diagrams}

\subsection{Set partitions and integer partitions}

The kind of Feynman diagram considered in the main text of this article is what might be
called a labeled Feynman diagram.
This appendix is a brief description of the passage from labeled Feynman diagrams to abstract Feynman diagrams,
that is, to isomorphism classes of Feynman diagrams.
This will help make a comparison with other approaches, such as that of \cite{A}
and \cite{E}. For the sake of exposition, the basic definitions will be repeated, so
this appendix will be largely self-contained. It starts with some combinatorial
background.

Let $U$ be a set with $n$ elements. A \emph{set partition} $\Gamma$ of $U$ is a collection
of non-empty subsets with no overlaps and whose union is $U$. Each subset $V$ in $\Gamma$ is called a
\emph{block}. The number of partitions of $U$ with $v$ blocks is the \emph{Stirling set partition number}
$S(n,v)$ (also known as the Stirling number of the second kind). The total number of partitions of $U$
is the \emph{Bell number} $B(n)$.

Let $n \geq 0$ be a natural number. An \emph{integer partition} of $n$ is a multiset of natural numbers $k \geq 1$
with sum $n$. Each such number $k$ is called a \emph{part} of $n$.
An integer partition is naturally specified by its \emph{type}: a sequence of numbers $p_k$ for $k = 1,2,3,\ldots$
such that
\begin{equation}
1 p_1 + 2 p_2 + 3 p_3 + \cdots + k p_k + \cdots = n.
\end{equation}
Thus $p_k$ represents the number of times the number $k \geq 1$ is used in the sum that gives $n$.
If the integer partition has $v$ parts, then
\begin{equation}
 p_1 +  p_2 +  p_3 + \cdots +  p_k + \cdots = v.
\end{equation}
The number of integer partitions of $n$ into $v$ parts is denoted $P(n,v)$. The total number of integer
partitions of $n$ is denoted $P(n)$.

For each set partition $\Gamma$ there is corresponding integer partition type $p$. As we shall see, the
number of set partitions with type $p$ is
\begin{equation}
O(p) = \frac{n!}{\prod_k p_k! (k!)^{p_k}}.
\end{equation}
It follows that the Stirling set partition number $S(n,v)$ is given
by
\begin{equation}
S(n,v) = \sum_p O(p),
\end{equation}
where the sum is over types $p$ with $1 p_1 + 2 p_2 + 3 p_3 + \cdots + k p_k + \cdots = n$ and
$ p_1 +  p_2 +  p_3 + \cdots +  p_k + \cdots = v$. The number of terms in this sum is $P(n,v)$.

In the following sections these ideas are illustrated in detail for $n=4$ and $n=6$. Here is a
summary for quick reference. For $n=4$
the numbers $P(4,v)$ for $v = 1,2,3,4$ are $1,2,1,1$ with sum $P(4) = 5$. The corresponding numbers
$S(4,v)$ are  $1,7,6,1$ with sum $B(4) = 15$.
For $n=6$
the numbers $P(6,v)$ for $v = 1,2,3,4,5,6$ are $1,3,3,2,1,1$ with sum $P(4) = 11$. The corresponding numbers
$S(6,v)$ are  $1,31, 90,65,15,1$ with sum $B(6) = 203$.





\subsection{Feynman diagrams}

For simplicity the discussion will concentrate on vacuum Feynman diagrams;
the term Feynman diagram with no modifier will mean vacuum Feynman diagram.
However, at the end there will be a brief reference to the more general concept of truncated
Feynman diagram.


Fix a set $U$ with $n$ elements.
A \emph{Feynman diagram} with label set $U$ has vertices and lines. The vertex part consists of a set partition $\Gamma$ of $U$.
The line part consists of a perfect matching $\sigma$ of $U$. (A perfect matching is a
set partition into two-element subsets.) Thus $n$ must be even, and $U$ consists
of all possible end points of lines. The case $n=0$ is allowed; in that case both $\Gamma$ and $\sigma$ are
empty, and there is only one diagram. A Feynman diagram built over $U$ may be pictured as a set $U$ of points
collected into bags that correspond to the blocks of the partition, together with lines between pairs of
points given by the perfect matching.

Let $F$ be the set of Feynman diagrams built over $U$. The number of Feynman diagrams is the number of set partitions
times the number of perfect matchings. The number of set partitions is $B(n)$, the $n$th Bell number.
Perfect matchings are set partitions corresponding to the integer partition with
type given by $p_2 = n/2$ and all other
$p_k = 0$. The general formula for $O(p)$ gives the number of perfect matchings as
\begin{equation}
m_n = \frac{n!}{\left(\frac{n}{2}\right)!\, 2^\frac{n}{2}}.
\end{equation}

A Feynman diagram defines a graph, where the blocks of the set partition
are the vertices of the graph, and the lines of the perfect matching are the edges of the graph.
In other words, each block is shrunk to a single vertex.
If a block has $k$ points, then the
corresponding vertex  has degree $k$, that is, it can have $k$ edges attached. The graph
may have multiple edges joining a pair of vertices. Also, the graph may have loops, that is, edges from a vertex to itself.


The contribution of a Feynman graph is calculated from data associated with a modification
of a Gaussian process. The set $\Xc$ is the index set for a Gaussian process. For a finite
set $W$ consider a function $a: W \to \Xc$; this is an assignment of indices. For each
such index assignment there is a corresponding set of occupation numbers $N_a(x) = \# \{j \mid a_j = x \}$.
The process is determined by interaction vertex coefficients $S_a$ and
line covariance coefficients $C^a$. These only depend on $a$ through the occupation number
coefficients $N_a$. In the case of a line covariance coefficient $C^a$ the set $W$ has exactly two
points. Thus when $N_a(x) = 2$ this is the variance $C^{xx}$, while when $N_a(x) = N_a(y) = 1$
this is the covariance $C^{xy}$.


If $\gamma = (\Gamma,\sigma)$ is a Feynman diagram in $F$, and $a: U \to \Xc$ is given,
the contribution has two main factors.
The factor $S(a,\Gamma) = \prod_{V \in \Gamma} S_{a_V}$
is the product of interaction factors associated with the vertices. Here $a_V: V \to \Xc$
denotes the restriction of $a$ to $V \subseteq U$. If a vertex $V$
is of large enough degree, then such a factor is supposed to be zero (or at least small).
This means that the contribution (or the main contribution) is from graphs with limited
degree of the vertices.
The factor $C(a,\sigma) =\prod_{W \in \sigma}  C^{a_W}$ is
the product of the covariances of the Gaussian process over the
pairs in the perfect matching. In other words, it is a product of
factors associated with the lines. The total contribution is obtained
by taking product of these two factors and then summing over the
index assignments, giving
\begin{equation}
t(\gamma) = \sum_{a: U \to  \Xc} S(a,\Gamma) C(a,\sigma).
\end{equation}
The contribution to the partition function of  the Feynman diagrams with given $U$ (representing $n$ endpoints
of lines) is
\begin{equation}
Z_n = \frac{1}{n!} \sum_{\gamma \in F} t(\gamma).
\end{equation}
The purpose of the discussion in the following sections is to relate this quantity to abstract
Feynman diagrams, defined as  equivalence classes of Feynman diagrams that capture the
graphical structure.

Before turning to examples, it may help to recall  graphical ideas associated with
Feynman diagrams on a set $U$. First, it makes sense to say that a diagram is connected.
This holds if $U$ is non-empty and cannot be partitioned in a way that decomposes the diagram into a pair of diagrams.
A stronger condition is to say that the diagram is line-irreducible. This says that the
diagram is connected and furthermore there
is no bridge in the diagram, that is no line whose removal disconnects the diagram.


Here are the two one-line ($n=2$) abstract Feynman diagrams.
\begin{itemize}
\item One vertex. Loop.
\item Two vertices. Bridge.
\end{itemize}
These are both connected, but the only line-irreducible diagram is the loop.


Here are the four connected two-line ($n=4)$ abstract Feynman diagrams $g$ and the corresponding number
$O(g)$ of labeled diagrams.

\begin{minipage}{\textwidth}
\begin{itemize}
\item One vertex. Double loop. $O(g) = 3$
\item Two vertices. Bridge-loop. $O(g) = 12$
\item Two vertices. Double parallel lines. $O(g) = 6$
\item Three vertices. Bridge-bridge. $O(g) = 12$
\end{itemize}
\end{minipage}

The only ones that are line-irreducible are the double loop and the double parallel lines.
Each abstract Feynman diagram comes from
a number of different labeled Feynman diagrams. To see how this works, start with a label set with four elements.
First pick the  set partitions that define a suitable vertex set. Then
find the perfect matchings that create
the lines of the appropriate diagram. The results for the four diagrams are
$1 \cdot 3 = 3$, $4 \cdot 3 = 12$, $3 \cdot 2 = 6$ and $6 \cdot 2 = 12$. This gives the statistics in the above list.



\subsection{Vertex sets}

As a preliminary step,  examine the vertex structure alone.
Let $P$ be the collection of set partitions of $U$. The permutation group $G$ of $U$ is a group of
order $n!$. It acts in a natural way on $P$. Let $\Pc = P/G$ be the collection of orbits of this action.
Each orbit $p$ is characterized by numbers $p_k$, where a set partition $\Gamma$ in $p$ has
$p_k$ blocks of cardinality $k$. These numbers specify an {integer partition} of the integer $n\geq 0$ into parts, where there
are $p_k \geq 0$ parts of size $k\geq 1$. Furthermore, if $\Gamma$ has $v$ blocks, then the integer
partition is into $v$ parts.
In the Feynman diagram context an integer partition is
considered as a \emph{vertex set}, where the
number of vertices that accommodate $k$ lines is $p_k$. Each vertex in the vertex set is
pictured as a point with $k$ half-lines emerging from it. The vertex set itself is a collection
of such vertices. A vertex set also could be called an \emph{abstract pre-Feynman diagram}, since it is related to
a similar concept introduced in \cite{A}. When the half-lines are paired up and joined to create a graph,
the result is an abstract Feynman diagram. The total number of vertices in a vertex set
is $v$. The number of half-lines emerging from
the vertices is the number $n$ of potential end points
of lines.


Consider a vertex set $p$ in $\Pc$ and set partition $\Gamma$ in $p$. Let $G_\Gamma$ be the
subgroup of $G$ that leaves $\Gamma$ invariant. This is the stabilizer (isotropy subgroup) of
 $\Gamma$. If $\Gamma$ has $p_k$ blocks of size $k$, then the order of this group is
\begin{equation}
A(p) = |G_\Gamma| = \prod_k p_k! \, (k!)^{p_k}.
\end{equation}
 This is because for each fixed $k$ one can
 permute the $p_k$ vertices, and within each vertex one can permute the $k$ lines that
emanate from this vertex.


If $\Gamma$ belongs to $p$ in $\Pc$, then $p = G \Gamma$ is the orbit of $\Gamma$ under the
group $G$. There is a natural map from $G/G_\Gamma$ to $G \Gamma$. In particular, the
size of the orbit $O(p) = |G \Gamma| $
is the number of cosets in $G/G_\Gamma$, which in turn
is
\begin{equation}
O(p) = |G \Gamma| = \frac{|G|}{|G_\Gamma|} = \frac{n!}{A(p)}.
\end{equation}
In other words, there are $O(p)$ set partitions for each integer partition $p$.


Consider the case $n=4$. There are five vertex sets (integer partitions of 4).
The total number of set partitions is $B_4 = 1 + (4 + 3) + 6 + 1 = 15$.  If we group according
to the number of blocks in the set partition, this is the expression for the Bell
number $B_4 = 1 + 7 + 6 + 1 = 15$ as a sum of Stirling set partition numbers. The statistics of the corresponding
five vertex sets are shown in Figure~1. The sum in the $O(p)$ column is of course $15$.
\begin{table}
\begin{tabular}{r|rrrr|rr}
$v$ & $p_1$ & $p_2$ & $p_3$ & $p_4$ & $A(p)$ & $O(p)$ \\ \hline
1& 0 & 0 & 0 & 1 & 24 & 1 \\
2& 1 & 0 & 1 & 0 & 6 & 4 \\
2&0 & 2 & 0 & 0 & 8 & 3 \\
3&2 & 1 & 0 & 0 & 4 & 6 \\
4& 4 & 0 & 0 & 0 & 24 & 1 \\
\end{tabular}
\vskip10pt
\caption{Statistics of two-line vertex sets}
\end{table}

For the case $n=6$ there are 11 different vertex sets (integer partitions of 6).
The total
number of set partitions is $B_6 = 1 + (6 + 15 + 10) + (15 + 60 +15) + (45 + 20) + 15 + 1 = 203$.
 If we group according
to the number of blocks in the set partition, this is the expression for the Bell
number $B_6 = 1 + 31 + 90 + 65 + 15 + 1 = 203$ as a sum of Stirling set partition numbers.
The statistics are given in Figure~2. The sum in the $O(p)$ column is 203.
\begin{table}
\begin{tabular}{r|rrrrrr|rr}
$v$ & $p_1$ & $p_2$ & $p_3$ & $p_4$ & $p_5$ & $p_6$ & $A(p)$ & $O(p)$  \\ \hline
1&0 & 0 & 0 & 0 & 0 & 1 & 720 & 1 \\
2&1 & 0 & 0 & 0 & 1 & 0 & 120 & 6\\
2&0 & 1 & 0 & 1 & 0 & 0 & 48 & 15 \\
2&0 & 0 & 2 & 0 & 0 & 0 & 72 & 10 \\
3&2 & 0 & 0 & 1 & 0 & 0 & 48 & 15 \\
3&1 &1 & 1 & 0 & 0 & 0 & 12 & 60 \\
3&0 & 3 & 0 & 0 & 0 & 0 & 48 & 15 \\
4&2 & 2 & 0 & 0 & 0 & 0 & 16 & 45 \\
4&3 & 0 & 1 & 0 & 0 & 0 & 36 & 20 \\
5&4 & 1 & 0 & 0 & 0 & 0 & 48 & 15 \\
6&6 & 0 & 0 & 0 & 0 & 0 & 720 & 1 \\
\end{tabular}
\vskip10pt
\caption{Statistics of three-line vertex sets}
\end{table}



\subsection{Abstract Feynman diagrams}




A Feynman diagram $\gamma$ on the label set $U$ (endpoints of lines) consists of
a set partition $\Gamma$ of $U$ (a collection of vertices) and a perfect matching $\sigma$ of $U$ (a collection
of lines). The permutation group $G$ of $U$
acts in a natural way on the set $F$ of Feynman diagrams on $U$.
Let $\Fc = F /G$ be the collection of orbits of this action. An element of $\Fc$ is called an
\emph{abstract Feynman diagram}.


The values of $t(\gamma)$ are constant for $\gamma$
belonging to a particular orbit $g$ in $\Fc$. It is thus consistent to denote the contribution of a diagram
in the orbit $g$ by $t(g)$. Let $O(g)$ be the number of points in the orbit of the abstract Feynman
diagram $g$, that is, the number of labeled Feynman diagrams with the same graphical structure.
In this notation we have
\begin{equation}
Z_n = \frac{1}{n!} \sum_{g \in \Fc} O(g) t(g).
\end{equation}
This expresses $Z_n$ as a sum over abstract Feynman diagrams of corresponding contributions.

Consider an abstract Feynman diagram $g$ in $\Fc$ and diagram $\gamma$ in $g$. Let $G_\gamma$ be the
subgroup of the permutation group $G$ on $U$ that leaves $\gamma$ invariant. This is the stabilizer (isotropy subgroup) of the
diagram $\gamma$. If $\gamma,\gamma'$ belong to the same orbit $g$, then $G_\gamma$ and
$G_{\gamma'}$ are conjugate. The isomorphism class of such stabilizers constitutes the abstract
\emph{automorphism group} of the abstract Feynman diagram $g$. Let $A(g) = |G_\gamma|$ for $\gamma \in g$ be
the order of this group.

If $\gamma$ belongs to $g$ in $\Fc$, then $g = G \gamma$ is the orbit of $\gamma$ under the
group $G$. There is a natural map from $G/G_\gamma$ to $G \gamma$. In particular, the
size of the orbit
\begin{equation}
O(g) = |G \gamma|
\end{equation}
 is the number of cosets in $G/G_\gamma$, which in turn
is $|G|/|G_\gamma| = n!/A(g)$. This gives the result
\begin{equation}
A(g) = |G_\gamma| = \frac{n!}{|G\gamma|} =  \frac{n!}{O(g)}.
\end{equation}
This leads to the remarkable identity
\begin{equation}
Z_n =  \sum_{g \in \Fc} \frac{1}{A(g)} t(g).
\end{equation}
This identity says that $Z_n$ is the sum of the weights of the
abstract Feynman graphs divided by their symmetry factors.

It is easy to pass from an abstract Feynman diagram to a Feynman diagram; all that
is required is to label the end points of the lines. Then an automorphism of this
labeled diagram is  a map from the label set to itself such that lines are taken
to lines and vertices are taken to vertices. A more abstract point of view is to
think of a map from end points of lines to end points of lines such that lines are taken
to lines and vertices are taken to vertices.

Example: Consider the abstract Feynman diagram $g$ with two vertices and two parallel lines.
The two vertices may be labeled with 1,2 and with 3,4 in such a way that one line
runs from 1 to 3 and the other from 2 to 4. The stabilizer group of the diagram is of order $A(g) = 4$.
There is the identity automorphism, the automorphism $(12)(34)$, the automorphism
$(13)(24)$, and the automorphism $(14)(23)$. From the abstract point of view
these do nothing, interchange the lines, interchange the vertices, and interchange the
vertices and lines.

In this same example, one can compute the orbit of this diagram. These are the other
diagrams with the same abstract structure that are obtained by permuting the end points
of lines. The number of such diagrams is $O(g) = 6$. There are three possibilities for
grouping the end points into vertices: 12 and 34, 13 and 24, 14 and 23. For each of these vertices one can join
the vertices with lines in two different ways. So the number of such diagrams is $O(g) = 3 \cdot 2 = 6$.

Example: The automorphism group acts on vertices, and it also acts on lines.
However  behind it all the automorphism group acts
on the end points of the lines. An extreme example is the loop on one vertex. The automorphism
group swaps the end points, so it has order $A(g) = 2$. The same thing happens for the
double loop on one vertex. The automorphism group includes the operation of interchanging the loops,
but it also includes swapping the end points of the individual loops. Thus for this example $A(g) = 8$.

Another way of approaching the classification of abstract Feynman
diagrams is to fix the partition $\Gamma$ belonging
to the vertex set $p$.
The stabilizer group $G_\Gamma$ of $\Gamma$ is a group of
order $A(p) = \prod_k p_k! (k!)^{p_k}$. The stabilizer group of a Feynman diagram
$\gamma$ is a subgroup of this group. The quotient $G_\Gamma/G_\gamma $ is
the orbit $G_\Gamma \gamma$ of the perfect matchings under this smaller group. These
orbits are smaller and relatively easy to compute. They are just all the ways
of attaching lines to a given vertex set to make the desired diagram.
Define
\begin{equation}
O(g | p) = |G_\Gamma \gamma|.
\end{equation}
So this gives another way
of computing $A(g) = |G_\gamma|$, namely,
\begin{equation}
A(g)  = \frac{|G_\Gamma|}{|G_\Gamma \gamma|} = \frac{A(p)}{O(g| p)} .
\end{equation}
 In particular, $A(g)$ must divide $A(p)$, which
in turn divides $n!$. An alternative way of doing the
computation of $A(g) = n!/O(g)$ is to use
\begin{equation}
O(g) = O(p) O(g| p) .
\end{equation}


Example: It is illuminating to look at the example of the diagram with two vertices and
two parallel lines. The vertex set $p$ consists of the two vertices, each of degree 2.
Say that these vertices are labeled 12 and 34.
The vertex set stabilizer group is of order $A(p) = 8$. It switches the end points within
each vertex and also switches the vertices. These permutations also act on
the Feynman diagrams. Consider the Feynman diagram $g$ with a 13 and a 24 line. The stabilizer
of $g$ has order $A(g) = 4$. The orbit of this diagram under the vertex set stabilizer group
has only two diagrams: the original diagram and the one with lines switched, that is,
with the same vertices but with a 14 and a 23 line. This shows that $O(g|p) = 2$. Since
$O(p) = 3$, the product is $O(g) = 3\cdot 2 = 6$, as it should be.




\subsection{Two-line diagrams}

It helps to look at examples more systematically.
Here are the seven two-line ($n=4$) abstract Feynman diagrams, together with
their decompositions into connected components.
\begin{itemize}
\item One vertex. Double loop.
\item Two vertices. Bridge-loop.
\item Two vertices. Double parallel lines.
\item  Two vertices. Loop $+$ loop.
\item Three vertices. Bridge-bridge.
\item Three vertices. Bridge $+$ loop.
\item Four vertices. Bridge $+$ bridge.
\end{itemize}
Four of these are connected. Of these, the double loop and the
double parallel line diagrams are the only line-irreducible diagrams.

Since there are $B_4 = 15$ set partitions and $m_4 = 3$
perfect matchings, there are 45 Feynman diagrams. There are 7 abstract Feynman diagrams.
The total number of diagrams grouped by vertex type is $3 + 12 + (6+3) + (12 + 6) + 3 = 45$.
For each of the five vertex sets, the sum of the sizes of the corresponding orbits is 3.
The statistics for these abstract Feynman diagrams are displayed in Figure~3. The sum in
the $O(g|p)$ column is the number of vertex sets times the number of
perfect matchings, that is, $5 \cdot 3 = 15$. The sum in the $O(g)$ column is the number of set partitions
times the number of perfect matchings, which is $15 \cdot 3 = 45$.
\begin{table}
\begin{tabular}{r|rrrr|rrrrr}
$v$&$p_1$ & $p_2$ & $p_3$ & $p_4$ & $A(p)$ & $O(p)$ & $O(g | p)$ & $O(g)$ & $A(g)$ \\ \hline
1&0 & 0 & 0 & 1 & 24 & 1 & 3 & 3 & 8\\
2&1 & 0 & 1 & 0 & 6 & 4 & 3 & 12 & 2\\
2&0 & 2 & 0 & 0 & 8 & 3 & 2 & 6 & 4\\
2&0 & 2 & 0 & 0 & 8 & 3 & 1 & 3 & 8\\
3&2 & 1 & 0 & 0 & 4 & 6 & 2 & 12 & 2\\
3&2 & 1 & 0 & 0 & 4 & 6 & 1 & 6 & 4\\
4&4 & 0 & 0 & 0 & 24 & 1 & 3 & 3 & 8\\
\end{tabular}
\vskip10pt
\caption{Statistics of two-line abstract Feynman diagrams}
\end{table}

\subsection{Three-line diagrams}

A nice test of the theory is to classify all three-line ($n=6$) Feynman diagrams.
The number of perfect matchings is $m_6 = 15$, so the total number
of Feynman diagrams is $203\cdot 15 = 3045$. However they only lead to
23 three-line  abstract Feynman diagrams. Here is the list.
\begin{itemize}
\item One vertex. Triple loop.
\item Two vertices. Bridge-double loop.
\item Two vertices. Double parallel lines with loop.
\item Two vertices. Loop $+$ double loop.
\item Two vertices. Triple parallel lines.
\item Two vertices. Loop-bridge-loop.
\item Three vertices. Bridge-loop-bridge.
\item Three vertices. Bridge $+$ double loop.
\item Three vertices. Bridge-double parallel lines.
\item Three vertices. Bridge-bridge-loop.
\item Three vertices. Loop $+$ bridge-loop.
\item Three vertices. Triple line triangle.
\item Three vertices. Loop $+$ Double parallel lines.
\item Three vertices. Loop $+$ loop $+$ loop.
\item Four vertices. Bridge-bridge-bridge.
\item Four vertices. Bridge-bridge $+$ loop.
\item Four vertices. Bridge $+$ double parallel lines.
\item Four vertices. Bridge $+$ loop $+$ loop.
\item Four vertices. Triple bridge star.
\item Four vertices. Bridge $+$ bridge-loop.
\item Five vertices. Bridge $+$ bridge-bridge.
\item Five vertices. Bridge $+$ bridge $+$ loop.
\item Six vertices. Bridge $+$ bridge $+$ bridge.
\end{itemize}
The statistics of these 23 diagrams are summarized in Figure~4. The sum
in the $O(g|p)$ column is the number of vertex sets times the number of perfect
matchings: $11 \cdot 15 = 165$, and the sum in the $O(g)$ column is the number of
set partitions times the number of perfect matchings:
$203 \cdot 15 = 3045$.
\begin{table}
\begin{tabular}{r|rrrrrr|rrrrr}
$v$&$p_1$ & $p_2$ & $p_3$ & $p_4$ & $p_5$ & $p_6$ & $A(p)$ & $O(p)$ & $O(g|p)$ & $O(g)$ & $A(g)$  \\ \hline
1&0 & 0 & 0 & 0 & 0 & 1 & 720 & 1 &  15 &  15 & 48 \\
2&1 & 0 & 0 & 0 & 1 & 0 & 120 & 6&   15 & 90 & 8   \\
2&0 & 1 & 0 & 1 & 0 & 0 & 48 & 15 &   12 & 180 & 4  \\
2&0 & 1 & 0 & 1 & 0 & 0 & 48 & 15 &   3 & 45 & 16  \\
2&0 & 0 & 2 & 0 & 0 & 0 & 72 & 10 &   6 & 60 & 12  \\
2&0 & 0 & 2 & 0 & 0 & 0 & 72 & 10 &   9 & 90 & 8 \\
3&2 & 0 & 0 & 1 & 0 & 0 & 48 & 15 &   12 & 180 & 4  \\
3&2 & 0 & 0 & 1 & 0 & 0 & 48 & 15 &    3 & 45 & 16 \\
3& 1 &1 & 1 & 0 & 0 & 0 & 12 & 60 &    6 & 360 & 2 \\
3& 1&1 & 1 & 0 & 0 & 0 & 12 & 60 &    6 & 360 & 2 \\
3& 1 &1 & 1 & 0 & 0 & 0 & 12 & 60 &    3 & 180 & 4 \\
3&0 & 3 & 0 & 0 & 0 & 0 & 48 & 15 &   8 & 120 & 6 \\
3&0 & 3 & 0 & 0 & 0 & 0 & 48 & 15 &    6 & 90 & 8 \\
3&0 & 3 & 0 & 0 & 0 & 0 & 48 & 15 &    1 & 15 & 48  \\
4&2 & 2 & 0 & 0 & 0 & 0 & 16 & 45 &    8 & 360 & 2 \\
4&2 & 2 & 0 & 0 & 0 & 0 & 16 & 45 &     4 & 180 & 4\\
4&2 & 2 & 0 & 0 & 0 & 0 & 16 & 45 &    2 & 90 & 8\\
4&2 & 2 & 0 & 0 & 0 & 0 & 16 & 45 &   1 & 45 & 16  \\
4&3 & 0 & 1 & 0 & 0 & 0 & 36 & 20 &    6 & 120 & 6\\
4&3 & 0 & 1 & 0 & 0 & 0 & 36 & 20 &    9 & 180 & 4 \\
5&4 & 1 & 0 & 0 & 0 & 0 & 48 & 15 &    12 & 180 & 4\\
5&4 & 1 & 0 & 0 & 0 & 0 & 48 & 15 &     3 & 45 & 16\\
6&6 & 0 & 0 & 0 & 0 & 0 & 720 & 1 &    15 & 15 & 48\\
\end{tabular}
\vskip10pt
\caption{Statistics of three-line abstract Feynman diagrams}
\end{table}

\subsection{Line-irreducible graphs}

Here are remarks on the further decomposition of Feynman diagrams.
For this purpose a  more general concept is needed.
An amputated Feynman diagram on $U$ consists of a set partition $\Gamma$ of $U$
and a perfect matching $\sigma$ of a subset of $U$. Again the blocks of the
set partition are regarded as vertices, and the matching pairs are regarded as
lines that run between vertices. The unmatched points in $U$ are each regarded
as a potential end point of a line---an amputated line.
A connected Feynman diagram is line-irreducible if it has no bridge, that is,
it has no line whose removal disconnects the diagram. When there are bridges,
their removal produces components that are line-irreducible amputated
diagrams.


There are only two one-line abstract Feynman diagrams with $n=2$. The one-vertex loop is line-irreducible, while
the two-vertex bridge decomposes into two $n=1$ diagrams each consisting of a vertex with an amputated line.
The four connected two-line abstract Feynman diagrams with $n=4$ provide a richer variety of
examples. The one-vertex double loop diagram is line-irreducible. The two-vertex bridge-loop diagram decomposes into
a $n=1$ vertex with amputated line diagram plus a $n=3$ vertex with loop and amputated line diagram. The two-vertex double parallel line diagram is line-irreducible. The three-vertex bridge-bridge diagram decomposes into three parts: two
$n=1$ vertex with amputated line diagrams and one $n=2$ vertex with double amputated lines diagram.




\subsection{Related work}

The article of Abdesselam \cite{A} is in a  somewhat different setting. His Feynman diagrams have oriented lines,
which corresponds in physics to bosons with charge. He uses
the Feynman diagram formulation to represent various kinds of functional composition, and this is
directional by nature. Nevertheless, there are elements in common with the present treatment.
The principal difference is that in the main part of the present treatment the species of Feynman diagrams is formulated as a functor on
colored sets. This means that the weight of a Feynman diagram does not have a sum over colors; that
sum arises only when the exponential generating function is computed. By contrast, in the treatment
by Abdesselam the species consists of abstract Feynman diagrams. These are regarded
as functors on sets, and the weight of an abstract Feynman diagram involves a sum.
The transition from one point of view to the other was presented earlier in this appendix.
The article \cite{A} has a rich variety of other combinatorial ideas, including a  symbolic
calculus of functional integration.


The Brydges--Leroux article \cite{BL} does not consider Feynman diagrams; the subject is
ordinary graphs. In the statistical mechanics of particles these would be called Mayer graphs. As
is well-known \cite{L}, there is a decomposition of connected graphs into 2-vertex-connected
subgraphs that is important for statistical mechanics.
In their article the graphs are defined over sets, so color does not play a role. The
interesting feature is that the authors study the decomposition of connected graphs into
2-edge-connected subgraphs. This is a close analog to what is done in the decomposition
of Feynman diagrams into line-irreducible graphs. It is not clear to the present author whether
this decomposition could also play a role in statistical mechanics.

Another fascinating direction is higher Legendre transforms. This subject is treated in the
articles of Cooper, Feldman, and Rosen (see for instance \cite{CFR}) and in the references they cite.
There is also a book by Vasiliev \cite{V} that gives a systematic account.


\section*{Acknowledgments}

This work was inspired by the writings of  Pierre Leroux  on the relation of combinatorics to
statistical physics, in particular by \cite{L,BL}. The author is grateful
to David Brydges for patient explanations of this subject. He thanks the Pacific
Institute for the Mathematical Sciences at the University of British Columbia for
providing a stimulating work environment. Finally, he wishes to acknowledge the
remarkable care and insight provided by the referee.

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\end{thebibliography}
\end{document}
