Abstract
We prove a multivariate LagrangeGood formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.
Introduction
Consider a power series of the form
In a previous paper [23], we have expressed the inverse map \(\zeta (\nu )\) (that is \(\rho (z) = \nu \) if and only if \(z = \zeta (\nu )\)) with the help of a tree generating functional. The main focus is to formulate these results for uncountably many colors. An uncountable color palette is completely natural if one studies thermodynamic functionals for inhomogeneous systems in order to derive variational or PDE formulations in different areas, for example in classical density function theory, liquid crystals, heterogenous materials, colloid systems, system of molecules with various shapes or other internal degrees of freedom. This is because “colors” correspond to positions or other continuous degrees of freedom (e.g., orientations).
If \(\rho \) and z are taking only nonnegative values, as in the above mentioned applications, then the form (1.1) is natural.
For a single complex variable \(z\in \mathbb {C}\), the inverse reads
where \(T^\circ (\nu )\) solves
and one recognizes the generating function for weighted rooted trees whose root is a ghost (that is, the root does not come with powers of \(\nu \) in the generating function).
On the other hand, as is wellknown, the coefficients of \(\nu \) in the inverse map \(\zeta (\nu )\) (and any functions of \(z=\zeta (\nu )\)) can be expressed in terms of the Lagrange inversion formula. Formally,
An additional integration by parts yields the more frequently encountered form of the Lagrange inversion formula
see e.g., [15, Appendix A6] or the recent survey [18]. The multivariate case is similar, with the determinant of a Jacobi matrix instead of the derivative \(\rho '(z)\). There is a variety of multivariate forms, see [17, 20, 29] and [4, 10, 26] for infinitely (countably) many variables.
Despite the large literature on LagrangeGood inversion and the increasing interest it attracts from combinatorialists, to the best of our knowledge no analogue for uncountably many variables has been considered up to now. Exception are [12, 28], where combinatorial identities are generalized to an arbitrary number of variables; however, the concept of summability by Mendéz and Nava is a restriction which would not allow to treat the abovementioned applications. The first aim of the present note is to fill this gap: we propose a multivariate LagrangeGood formula for functionals of uncountably many variables (Theorem 3.1).
The second aim is to clarify the relation with the tree formula from Proposition 2.6 in [23]. Just as Gessel’s proof of the LagrangeGood inversion formula for finitely many variables, our proof starts from a representation of the inverse in terms of trees. In contrast, determinants are associated with digraphs that may have cycles; equality arises because of cancellations as clarified in Proposition 3.4. There is actually more to this: when interested in proving asymptotic formulas and checking the validity (absolute convergence) of the power series, the tree formula is easier to handle. In fact, in [26], we had to show that the determinant was bounded and then in [23], we realized that the determinant is not actually there due to the cancellations and therefore we deduced better bounds with less effort. The observation that determinants might be a hindrance to asymptotic analysis is also behind determinantfree, socalled arborescent forms of LagrangeGood, see [5, 8, 19].
It is noteworthy that Good’s original motivation [20, 21] for generalizing Lagrange’s inversion formula was the enumeration of various kinds of trees, thus starting from the functional equation satisfied by combinatorial generating functions. The antipodal view point is to start from the inversion problem, consisting in solving a given functional equation; then one has two options: either apply one of the versions of the LagrangeGood inversion formula, or derive an expression for the solution directly with trees. Solving inversion problems with different types of trees is common practice in many areas: Butcher series in numerics, Gallavotti trees in RG group, Lindstedt series in KAM theory [16], algebra [6, 34], see Sect. 5 for more details and a discussion of the relation to the approach taken in this paper.
The article is organized as follows. Section 2 recalls the multivariate formulas that we seek to generalize and introduces formal power series associated with uncountable spaces. In Sect. 3 we set up the problem and present the main result, which is then proven in Sect. 4. We draw connections to other inversion formulas based on trees in Sect. 5. “Appendix A” recalls some formulas for formal power series. For our proof, we propose a formalism of colored species with uncountable color space, which allows us to adapt the proof given by Gessel [17] for finitely many variables to the case of uncountably many variables; this formalism is described in detail in “Appendix B”.
Preliminaries
Multivariate Lagrange inversion
Here, we recall the form of the Lagrange inversion formula for formal power series in finitely many variables \(z_1,\ldots , z_d\in \mathbb C\) that we seek to generalize. For \(\varvec{n}= (n_1,\ldots , n_d) \in \mathbb {N}_0^d\) we write \(\varvec{z}^{\varvec{n}} = z_1^{n_1}\cdots z_d^{n_d}\) and \([\varvec{z}^{\varvec{n}}] F(\varvec{z})\) denotes the coefficient of the monomial \(\varvec{z}^{\varvec{n}}\) in the series F, i.e., if \(F(\varvec{z})= \sum _{\varvec{n}} f_{\varvec{n}} \varvec{z}^{\varvec{n}}\), then \([\varvec{z}^{\varvec{n}}] F(\varvec{z}) = f_{\varvec{n}}\).
Suppose, we are given a family \((A_i(z_1,\ldots , z_d))_{i=1,\ldots ,d}\) of formal power series whose coefficient of order zero vanishes, \(A_i(\varvec{0}) =0\). Define additional formal power series^{Footnote 1}\(\rho _1,\ldots , \rho _d\) by
There is a uniquely defined family of power series \((\zeta _1,\ldots , \zeta _d)\) such that
as an equality of formal power series, furthermore \((\zeta _1,\ldots , \zeta _d)\) also satisfies
as an equality of formal power series in the variables \(\nu _1,\ldots , \nu _d\). Let \(\Phi (z_1,\ldots ,z_d)\) be yet another formal power series. Then,
Equation (2.4) follows from [17, Eq. (4.5)], see also Theorem 8(b) in [7, Chapter 3.2]. Variants for countably many variables are available [4, 10].
Specializing to \(\Phi (\varvec{z}) = z_k\) with \(k\in \{1,\ldots , d\}\), we obtain
compare (1.4) for \(d=1\). Conversely, Eq. (2.4) easily follows from Eq. (2.5) as well. Equation (2.5) allows us to express the coefficients of the unknown reverse series \(\zeta _i(\varvec{\nu })\) in terms of coefficients of the known series \(A_i(\varvec{z})\) and functions thereof. Equation (2.5) should be contrasted with tree formulas derived from the following functional equation (combining (2.1) and (2.3) )
see also Sect. 5.
Our goal is to generalize Eqs. (2.4) and (2.5) to a situation where the finite set \(\{1,\ldots , d\}\) is replaced with a possibly uncountable space \({\mathbb {X}}\), and to provide adequate combinatorial interpretations. The tree formula for the inverse in uncountable color space was already proven in [23], it is recalled in Sect. 4.2.
Formal power series
Let us briefly motivate the definition of formal power series adopted in [23, Appendix A]. A formal power series in finitely many variables \(z_1,\ldots , z_d\) may be written as
for some suitable family of coefficients \(a(\varvec{n})\), where \(\varvec{n} ! =n_1! \ldots n_d!\) but also as
with coefficients \(f_n(i_1,\ldots , i_n)\) that are invariant under permutation of the argument, where \( a(\varvec{n}) = f_n(i_1,\ldots , i_n)\), whenever \( \#\{ i_1, \ldots , i_d \, :\, i_n =j\} = n_j\) for all \(j\in \{1 , \ldots ,d\}\) and \(n = n_1 + \ldots + n_d\). Eq. (2.8) is less elegant than the multiindex formula (2.7) but redeems itself by a straightforward generalization to uncountable spaces. (Other benefits of Eq. (2.8) are discussed by Abdesselam in the context of Feynman diagrams and tensors [1, Sect. III].) If \(\{1,\ldots , d\}\) is replaced with a measurable space \(({\mathbb {X}},{\mathcal {X}})\) it is natural to replace sums with integrals and switch from a vector \((z_1,\ldots , z_d)\) to a measure \(z(\mathrm {d}x)\). Accordingly, the power series we are interested in are of the form
the coefficients consist of a scalar \(f_0\in \mathbb {C}\) and symmetric measurable functions \(f_n: {\mathbb {X}}^n\rightarrow \mathbb {C}\), \(n\in \mathbb {N}\). As usual for formal power series, we do not want to deal with questions of convergence and downgrade (2.9) to a mnemotechnic notation for the sequence \((f_n)_{n\in \mathbb {N}_0}\). We also want to define function and measurevalued formal power series F(q; z), \(K(\mathrm {d}q;z)\).
Definition 2.1
Let \(({\mathbb {X}}, {\mathcal {X}})\) be a measurable space.

(a)
A (scalar) formal power series on \({\mathbb {X}}\) is a family \((f_n)_{n\in \mathbb {N}_0}\) consisting of a scalar \(f_0\in {\mathbb {C}}\) and symmetric measurable functions \(f_n:{\mathbb {X}}^n\rightarrow \mathbb {C}\).

(b)
A functionvalued formal power series is a family \((f_n)_{n\in \mathbb {N}_0}\) of measurable maps \(f_n:{\mathbb {X}}\times {\mathbb {X}}^n\rightarrow \mathbb {C}\) such that \((q,x_1,\ldots , x_n)\mapsto f_n(q;x_1,\ldots , x_n)\) is symmetric in the \(x_j\) variables.

(c)
A measurevalued formal power series is a family \((k_n(\mathrm {d}q;x_1,\ldots , x_n))_{n\in \mathbb {N}_0}\) consisting of a measure \(k_0(\mathrm {d}q)\) on \({\mathbb {X}}\) and kernels \(k_n: {\mathcal {X}}\times {\mathbb {X}}^n\rightarrow \mathbb {R}_+\) such that for each \(B\in {\mathcal {X}}\), the map \(k_n(B;\cdot )\) is symmetric in the \(x_j\) variables.
Definition 2.1(b) and (c) are formulated for functions of a single variable \(q\in {\mathbb {X}}\) and nonnegative measures, they extend in a straightforward way to functions of several variables or complex measures. Standard operations such as sums and products are defined in “Appendix A.” The operations turn the set of scalar formal power series into an algebra that corresponds to the algebra of symmetric functions from Ruelle [31, Chapter 4.4].
Example 2.2
(Exponential). Let \(\varphi \) be a nonnegative measurable function on \(\mathbb {X}\), then the power series associated with the following exponential is
Example 2.3
(Monomials). Let \(m\in \mathbb {N}\) and \(q_1,\ldots , q_m\in {\mathbb {X}}\). Let \(\delta _{p,q}\) be the Kronecker delta, equal to 1 if \(p = q\) and 0 if \(p\ne q\). Then for every measure z on \({\mathbb {X}}\), we have
which is of the form (2.9) with \(f_n \equiv 0\) for all \(n\ne m\).
Example 2.4
(The measure \(z(\mathrm {d}q)\)). Let \(z(\mathrm {d}q)\) be a measure on \(({\mathbb {X}}, {\mathcal {X}})\). Then for all \(B\in {\mathcal {X}}\),
with kernel \(k_1(B;x)=\delta _{x}(B)\), i.e., \(k_1(\mathrm {d}q;x)\) is the Dirac measure at x. Accordingly, we may view \(z(\mathrm {d}q)\) as a measurevalued formal power series in the sense of Definition 2.1(c), with \(k_n\equiv 0\) for all \(n\ne 1\) and \(k_1(\mathrm {d}q; x) = \delta _x(\mathrm {d}q)\). The measure \(z(\mathrm {d}q)\) replaces the set of monomials \((z_i)_{i=1,\ldots , d}\) that appear naturally for power series of finitely many variables, with z(B) the analogue of \(\sum _{i\in B} z_i\).
Variational Derivatives and Extraction of Coefficients
Just as for usual power series, coefficients can be extracted by taking derivatives at the origin. In our context, the correct notion of derivative is a variational derivative defined as follows.
Definition 2.5
Let \(F(z) = f_0 + \sum _{n=1}^\infty \frac{1}{n!} \int _{{\mathbb {X}}^n} f_n(x_1,\ldots , x_n) z^n(\mathrm {d}\varvec{x})\) be a formal power series, i.e., \((f_n)_{n\in \mathbb {N}_0}\) is a family of symmetric functions as in Definition 2.1(a). The variational derivative of order k is the functionvalued formal power series with coefficients
Thus,
In particular, \(f_k(q_1,\ldots , q_k;z)\) is equal to the term of order zero in the formal power series \(\frac{\delta ^k f}{\delta z^k}(\varvec{q};z)\). Below, we often use the notation
Definition 2.5 is motivated by the following formal computation. For small \(t\in \mathbb {R}\) and another measure \(\mu \) on \({\mathbb {X}}\), we have
and
Fredholm Determinant
Let \(K:{\mathbb {X}}\times {\mathbb {X}}\rightarrow \mathbb {R}\) be a kernel and \({\mathbb {K}}\) the associated integral operator in \(L^2({\mathbb {X}}, {\mathcal {X}}, z(\mathrm {d}x))\), given by
For sufficiently regular kernels K, the Fredholm determinant \(\det ( \mathrm {Id}  {\mathbb {K}})\) is
see e.g., Sect. 3.11 in [32]. The righthand side of (2.17) is always welldefined as a formal power series in z, without any regularity assumptions on the kernel. Accordingly, we adopt (2.17) as a definition of the Fredholm determinant on the level of formal power series.
The definition is easily extended to kernels \(K_z\) and associated operators \({\mathbb {K}}_z\) that are themselves given by formal power series, as in Eq. (3.5). That is, suppose we are given a family of measurable functions \(k_0:{\mathbb {X}}\times \mathbb X\rightarrow \mathbb {R}\) and \(k_n : {{\mathbb {X}}} \times {\mathbb {X}} \times {\mathbb X}^n \rightarrow \mathbb {C}\) such that \((q,q';x_1 ,\ldots x_n) \mapsto k_n(q,q';x_1 ,\ldots x_n)\) is symmetric in the \(x_i\) variables, and define
The determinant \(\mathrm {det}\, ((K_z(q_i,q_j))_{i,j=1,\ldots ,n})\) is a combination of products of power series, the integral of formal power series is defined using (A.6). Therefore, the \(n\times n\) determinants and integrals in (2.17) stay welldefined as a formal power series with \(K_z\) instead of K. For the nth summand on the righthand side of (2.17), the first nonzero term in the power series expansion has degree n, because of the “integration” with \(z^n(\mathrm {d}\varvec{x})\). Hence, the contributions for the term of degree \(m \in \mathbb {N}\) in the formal power series of \(K_z\) comes from summands on the righthand side for \(n \le m\). Therefore, the coefficient of the degree m is a finite sum of finite products of the functions \(k_n\), and hence the infinite series on the righthand side is rigorously defined as a formal power series.
We conclude with a remark that may be helpful for readers which are not too keen on working with Fredholm determinants.
Remark 2.6
One can replace the Fredholm determinant with determinants of finite matrices. Recall from Example 2.3 how to interpret \(z(\{q\})\) as a formal power series.
Let \(n\in \mathbb {N}\) and \((q_1,\ldots ,q_n)\in {\mathbb {X}}^n\). Set \(Q=\{q_i:\, i=1,\ldots ,n\}\). Then, the nth coefficient of the Fredholm determinant \(\det (\mathrm {Id} {\mathbb {K}}_z)\), evaluated at \((q_1,\ldots ,q_n)\), is equal to the nth coefficient at \((q_1,\ldots ,q_n)\) of the \((\#Q)\times (\#Q)\)matrix
One can replace Q by any \(Q' \supset Q\) without altering the coefficient
If one additionally wants to avoid the use of measures for z, because one is either only interested in densities or maybe even in generalized functions, then in order to compute the nth. coefficient it is sufficient to consider the following determinant
where m is a reference measure on \(\mathbb {X}\), for example typically the Lebesgue measure.
For both cases, more details can be found at the end of “Appendix A.” The use of the Fredholm determinant gives the most natural connection to the combinatorics. All other interpretations will give rise to the same result as the coefficients of the associated formal power series are unchanged.
Main Results
Let \(({\mathbb {X}}, {\mathcal {X}})\) be a measurable space and
a functionvalued formal power series in the sense of Definition 2.1(b). Thus, each \(A_n: {\mathbb {X}}\times \mathbb X^n\rightarrow \mathbb {C}\) is a measurable function that is symmetric in the \(x_j\)variables. Define a measurevalued formal power series \(\rho (\mathrm {d}q;z)\) by
The coefficients of the power series on the righthand side are defined rigorously by Eqs. (A.9) and (A.7). We would like to determine the inverse power series \(\zeta [\nu ](\mathrm {d}q)= \zeta (\mathrm {d}q;\nu )\), that is
with the composition defined by (A.13). In a previous article [23], we have proven that the inversion is always possible on the level of formal power series, and we gave sufficient conditions for the absolute convergence of the involved power series. Precisely, concerning the formal inverse, we have proven that there is a unique family of formal power series \((T_q^\circ )_{q\in {\mathbb {X}}}\) that solves the fixed point equation
compare [7, Theorem 3.2.2] for finitely many variables. Moreover, the measurevalued formal power series
(see again (A.7)) satisfies Eq. (3.3).
Further we have shown [JKT2019, Proposition 2.6] that the power series \(T_q^\circ \) is the generating function for rooted weighted trees whose root has color q and is a ghost. (It does not come with powers of z in the generating functional.) The tree formula for the inverse is recalled in detail in Sect. 4.2. Corollary 3.5 provides an alternative representation as the coefficient of another power series, generalizing the multivariate Lagrange inversion formula (2.5).
First, however, we generalize Eq. (2.4) to uncountably many colors. The determinant in Eq. (2.4) is replaced with a Fredholm determinant. Define the kernel
The variational derivative has been introduced in Definition 2.5, see also Eq. (A.4). In particular,
Consider the formal operator
which we may view as a (formal) integral operator in \(L^2(\mathbb X,{\mathcal {X}},z(\mathrm {d}q))\) with kernel \(K_z(q',q)\). The Fredholm determinant \(\mathrm {det}(\mathrm {Id}  {\mathbb {A}}_z)\) is defined, as a formal power series, as in Sect. 2.4, see also Eq. (3.9) below.
Theorem 3.1
(LagrangeGood inversion). Let \(\rho [z](\mathrm {d}q)= z(\mathrm {d}q) \exp \bigl (  A(q;z)\bigr )\), where A is defined in (3.1). We denote by \(\zeta [\nu ](\mathrm {d}q)\) the formal power series of the inverse of \(\rho [z](\mathrm {d}q)\). Let \(\Phi (z) = \Phi _0+ \sum _{n=1}^\infty \frac{1}{n!}\int _{{\mathbb {X}}^n} \Phi _n(\varvec{x}) z^n(\mathrm {d}\varvec{x})\) be a further formal power series.
Define a formal power series \(\Psi \) (using (A.13)) by
Then, for all \(n\in \mathbb {N}\) and \((q_1,\ldots ,q_n) \in {\mathbb {X}}^n\), \( \Psi _n(q_1,\ldots ,q_n)\) is equal to the term of order zero in the formal power series
By the definitions adopted in Sect. 2.4, the Fredholm determinant in (3.8) is given by
we will sometimes use the heuristic notation
Remark 3.2
Following Remark 2.6, the Fredholm determinant in Theorem 3.1 can be replaced by usual determinants in two ways:

(1)
Using restrictions, we get that
$$\begin{aligned} \det \left( \left( \delta _{qq'}  z(\{q\})\frac{\delta }{\delta z(q')} A(q;z)\right) _{q,q'\in Q}\right) , \end{aligned}$$(3.10)where
$$\begin{aligned} Q:=\{q_i \mid i =1,\ldots , n\}. \end{aligned}$$(3.11)The replacement is also possible as well if we use instead of Q any bigger set \(Q'\supset Q\).

(2)
If z has a density with respect to a reference measure m, which we denote also by z, or if z is a generalized function one can replace the Fredholm determinant by
$$\begin{aligned} \int _{\mathcal {X}}\mathrm {det}\left( \Bigl ( \delta (x_j  x_j)  z(x_i) \frac{\delta }{\delta z(x_j)} A(x_i;z) \Bigr )_{i,j = 1, \ldots , n}\right) m(\mathrm {d}x_1) \ldots m(\mathrm {d}x_n),\nonumber \\ \end{aligned}$$(3.12)where the expression is welldefined for nice enough A; indeed see (3.9).
The usual determinant may feel more elementary; however, the Fredholm determinant is defined as a series of usual determinants anyhow. If one wishes to apply the theorem in a functionalanalytic context the Fredholm determinant may make sense as an actual determinant of an operator, while \(z(\{q\})\) in the finite matrix, could always be zero in the function space considered, for example in \(L^p\)spaces.
Remark 3.3
We stress that the determinant in Eq. (3.10) is the determinant of a matrix indexed by a set of colors Q and not by indices \(i,j\in \{1,\ldots , n\}\). Crucially, if \(q_i = q_j\) for some \(i \ne j\), the color \(q= q_i = q_j\) gives rise to only one row and column in the matrix. As a consequence, colors may repeat among the \(q _i\)’s but the determinant still be nonzero.
This is consistent with the determinant in Eq. (2.4) for finitely many variables. In analytic proofs [20], the determinant comes in via a complex change of variables in a contour integral. In particular, every variable \(z_k\) appears only once in the determinant, even when we are interested in coefficients of monomials with powers \(n_k\ge 2\). See also Eq. (A.21).
When \(\Phi (z) =1\), we have \(\Psi _n\equiv 0\) for all \(n\ge 1\), and we obtain the following striking result.
Proposition 3.4
For all \(n\ge 1\) and \((q_1,\ldots ,q_n)\in {\mathbb {X}}^n\), the term of order zero in the formal power series
vanishes.
Note that the expression in the curly bracket also depends on n. Remark 3.2 applies here as well. These cancellations give rise to the inversion formula expressed in terms of trees as given in Sect. 4.2. Similar cancellations happen in [2] where the formal inverse is expressed as a onepoint correlation function \(\rho ^{(1)}(x)\) of an appropriately chosen complex Bosonic gas. In the latter case, cycles are canceled by the partition function Z leaving one tree with root color x.
Another special case of Theorem 3.1 corresponds to the choice \(\Phi (z) = z(B)\) with \(B\subset {\mathbb {X}}\) measurable. (The analogue for finitely many variables is \(\sum _{i\in B} z_i\).)
Corollary 3.5
(Determinant formula for the inverse). For every \(B\in {\mathbb {X}}\), the nth coefficient of \(\zeta (B;\nu )\), evaluated at \((q_1,\ldots , q_n) \in {\mathbb {X}}^n\), is equal to the nth coefficient of
evaluated at \((q_1,\ldots , q_n)\).
The proposition provides an analogue for (2.5).
LagrangeGood Inversion. Proof of Theorem 3.1
Enriched Maps, CycleRooted Trees, Weights
A key ingredient of the bijective proofs by Labelle and Gessel [17, 27] is a graphical representation of mappings. Let us briefly recall some standard vocabulary, following [7, Chapters 3.1 and 3.2]. Let V, S be disjoint finite set.
A partial endofunction on \(U:=V\cup S\) with domain V is a mapping \(f:V \rightarrow U=V \cup S\). Any partial endofunction is associated with a directed graph \(G=(U,E)\), or digraph for short, with vertex set U and directed edges \(E = \{ (v,f(v))\mid v\in U \}\subset U\times U\). The digraph G may have selfedges (v, v), called loops. Loops in G correspond to fixed points of the mapping, \(f(v) =v\). The graphs G obtained in this way have two types of vertices: Vertices \(v\in V\) have exactly one outgoing edge because f maps v to exactly one vertex f(v). Vertices \(v\in U\setminus V=S\) are sinks, i.e., they have no outgoing edge, because they do not belong to the domain of f.
In both cases the outdegree in G of every vertex is either 0 or 1. We call digraphs with this property functional digraphs. Clearly there is a onetoone correspondence between functional digraphs G and partial endofunctions f and we often identify them, cf. Fig. 1.
In order to take into account the exponential \(\exp ( A(q;z))\) in our formulas, we enrich partial endofunctions with an additional structure: To each element \(v\in V \cup S\), we add a set partition \( P_v\) of the elements of the preimage \(f^{1}(\{v\})\) (the preimage is often called fiber of v). This is a special case of the Renriched structures often used in combinatorial proofs of Lagrange inversion formula [7, Definitions 3.1.1 and 3.1.8]. It is customary to represent Rstructure as a structure on the incoming edges of the functional digraph; in our case, \( P_v\) is represented as a set partition of the incoming edges (w, v).
Definition 4.1
Let V and S be finite possibly empty disjoint sets. If V is empty, we set \(\mathcal {M}^S[V] = \mathcal {M}^S[\varnothing ] := \varnothing \). If V is nonempty, we define \(\mathcal {M}^S[V]\) to be the collection of pairs \({\bar{f}} = (f, (P_v)_{v\in V\cup S})\) such that

f is a map \(f:V\rightarrow V\cup S\).

For each \(v\in V\cup S\), \(P_v\) is a partition into nonempty sets of the preimage \(f^{1}(\{v\})\). If the preimage is empty we set \(P_v = \varnothing \).
An element \({\bar{f}}= (f,( P_v)_{v\in V\cup S}) \in {\mathcal {M}}^S[V]\) is called enriched partial endofunction on \(V\cup S\), with domain V and sink set S, or enriched map for short.
For a typical element of \(\mathcal {M}^S[V]\) see Fig. 2. To lighten notation, if \(S= \varnothing \), we drop the superscript S and write \({\mathcal {M}}[V]\). If \(S = \{\circ \}\) is a singleton, we drop the braces in the notation and write \(\mathcal M^\circ [V]\) instead of \(\mathcal {M}^{\{\circ \}}[V]\).
A digraph G is connected if any two distinct vertices v, w can be connected by a path from v to w or from w to v. The connected components of functional digraphs are of two types: trees and cyclerooted trees, by which we mean the following.
Definition 4.2
Let \(G= (V\cup S, E)\) be a functional digraph with domain V, that is a directed graph such that every vertex \(v\in V\) has outdegree 1 and every vertex \(w\in S\) has outdegree zero.

(a)
A cycle in G is a sequence \(v_1,\ldots , v_n\) in V such that \((v_i,v_{i+1})\in E\) for all \(i\in \{1,\ldots , n1\}\) and \((v_n,v_1) \in E\). This includes loops (\(n=1\)).

(b)
G is a vertexrooted tree if it is connected and it has no cycle. Then, S is necessarily a singleton \( =\{\circ \}\), the root is the unique sink \(\circ \).

(c)
G is a cyclerooted tree if it is connected and it has exactly one cycle. Then, S must be empty.The unique cycle in G is called the root cycle of G.
By some abuse of language, an enriched map or graph are called vertexrooted or cyclerooted tree if the underlying graph G is vertex or cyclerooted tree, respectively. The class of enriched maps \({\bar{f}}\in {\mathcal {M}}^\circ [V]\) that are vertexrooted trees is denoted \(\mathcal {T}^\circ (V)\).
Finally, we define weights of enriched maps. For \({\bar{f}} = (f,(P_v)_{v\in V\cup S})\in {\mathcal {M}}^S[V]\) and \(\varvec{x}\in \mathbb X^{V\cup S}\), the weight of \({\bar{f}}\), given the coloring \(\varvec{x}\), is
We denote for short \( {\varvec{x}}_W:=(x_w)_{w\in W}\). For a graphical representation, see Fig. 3.
The Tree Formula for the Inverse Power Series
For our further calculation, we need a combinatorial expression for the inverse power series. Such a representation can be obtained directly without using LagrangeGood type formulas. We briefly recall the tree formula for the inverse power series proven in [23]. Let
and
Lemma 4.3
The family \((T_q^\circ (\nu ))_{q\in {\mathbb {X}}}\) fulfills the following functional equation as formal power series
For finite spaces \({\mathbb {X}}\), the lemma follows from Theorem 2 in Chapter 3.2 on implicit species in the book by Bergeron, Labelle, and Leroux [7].
Proof
The lemma follows from [23, Lemma 2.1 and Proposition 2.6], we give a selfcontained proof for the reader’s convenience. Calling \(\{V_1, \ldots , V_m\}\) the partition \(P_\circ \) of the vertices incident to the root \(\circ \), we can rewrite \(t_n\) using the tree structure of the associated graph as
where we defined:
(Compare with the calculations (formulas (26)(28)) in the proof of Theorem 1 in [2]].) Note that according to (A.2) the expression \(B(q;V_i)\) is the \(\# V_i\)th. coefficient in powers of \(\nu \) of
According to (A.9), the righthand side of (4.5) is the nth coefficient of the exponential of (4.7). \(\square \)
When dealing with generating functions of rooted trees we have two choices, either look at trees rooted in a ghost or trees rooted in a labeled vertex. In the univariate case (no colors), these give rise to two different generating functions \(T^\circ (\nu )\) and \(T^\bullet (\nu )\) related by \(T^\bullet (\nu )= \nu T^\circ (\nu )\). The multivariate case (finitely many colors) gives rise to a relation of the type \( T_i^\bullet (\nu )= \nu _i T^\circ _i(\varvec{\nu }) \), with i the color of the root. This last relation should remind the reader of the relation \( \zeta (\mathrm {d}q;\nu ) = \nu (\mathrm {d}q) T_q^\circ (\nu ) \) used to define \(\zeta (\mathrm {d}q;\nu )\) in (3.4). As a consequence, we should expect that the measurevalued series \(\zeta (\mathrm {d}q;\nu )\) corresponds to rooted trees with the root integrated over.
The next lemma makes this statement precise. It says that the series \(\zeta (B;\nu )\) is given by a sum over rooted trees with root color in B.
Lemma 4.4
Let \(\zeta (\mathrm {d}q;z) = T_q^\circ (q;z) z(\mathrm {d}q)\). The nth coefficient of \(\zeta (B;\nu )\), evaluated at \((q_1,\ldots , q_n)\), is equal to
and the term of order zero vanishes, \(\zeta _0(B) =0\).
Proof
The lemma is an immediate consequence of the definition (3.4) of \(\zeta (\mathrm {d}q;\nu )\) and Eq. (A.7). \(\square \)
Theorem 4.5
For \(\rho \) given in (2.1) we have \((\zeta \circ \rho )(\mathrm {d}q;z) = z(\mathrm {d}q)\) and \( (\rho \circ \zeta )(\mathrm {d}q;\nu ) = \nu (\mathrm {d}q)\) as an equality of measurevalued formal power series, with the composition defined in (A.13).
Proof
(Proof sketch) By the definition of \(\rho (\mathrm {d}q; z) = \exp (A(q;z)) z(\mathrm {d}q)\) and Lemma 4.3, we have
The previous chain of equalities is formal but it can be rigorously justified by properties of operations on formal power series (e.g., associativity of the product), defined in terms of coefficients only; we leave the details to the reader. The equality \((\zeta \circ \rho )(\mathrm {d}q;z) = z(\mathrm {d}q)\) is proven by a similar argument and another fixed point equation, see [23, Lemma 2.12]. \(\square \)
CycleRooted Forests. Proof of Proposition 3.4
For the proof of Proposition 3.4, we relate the righthand side of Eq. (3.13) to a sum over cyclerooted forests and look for combinatorial cancellations. We start with the exponential. Let \(E_k^n(\varvec{q};\varvec{x})\) be the coefficients of the exponential appearing in Proposition 3.4, i.e.,
Figure 2 depicts a forest of vertexrooted and cyclerooted trees. The vertices \([n]\setminus I\) are represented as white.
Lemma 4.6
Let \(E^n_k\) be as in (4.10). For \(n\in \mathbb {N}\), \(\varvec{q}\in {\mathbb {X}}^n\), and \(I\subset [n]\), we have
Notice that Lemma 4.6 does not address general coefficients \(E^n_{\#I}(q_1,\ldots , q_n;(x_i)_{i\in I})\), but only the special case \(x_i = q_i\). Roughly, \(E_{\#I}^n(q_1,\ldots , q_n; (x_i)_{i\in I})\) is a sum over enriched maps from I to \(\{1,\ldots , n\}\), but if \(x_i \ne q_i\) there are colorconflicts and the colordependent weight of a map is inappropriate to give a representation.
Proof
By \(\exp \left( \sum _{i=1}^n A(q_i;z)\right) = \prod _{i=1}^n \exp \left( A(q_i;z)\right) \) and (A.3), we can write for all \(\varvec{x}\in {\mathbb {X}}^I\)
where \(I_1,\ldots ,I_n\) are pairwise disjoint subsets, with \(I_k = \varnothing \) allowed, whose union is I and \({\varvec{x}}_I = (x_i)_{i \in I}\). By (A.8), we have that:
where \(\mathcal {P}(I_k)\) is the collection of set partitions \(\{W_1,\ldots ,W_r\}\) of \(I_k\) into nonempty sets \(W_i\). Combining the two, we obtain that
Each ntuple \((I_1,\ldots ,I_n)\) gives rise to a map \(f:I\rightarrow [n]\) by defining \(f^{1}(\{k\}) = I_k\); the correspondence is clearly onetoone. Considering \({\bar{f}} = \left( f , (P_k)_{k \in [n]} \right) \) and \(\varvec{x}_I=\varvec{q}_I\) we recognize the weight \(w(\bar{f},\varvec{q})\) and obtain (4.11). \(\square \)
Next, we turn to the determinant on the righthand side of Eq. (3.13). Write
where \( Q= \{q_i\mid i=1,\ldots , n\}\) is defined as in (3.11). We start with the interpretation of the lefthand side as a formal power series, cf. (3.9),
and next we use the combinatorial interpretation of the matrix element
which by Eq. (3.5) takes the form
In \(M_{p,p'}(z)\), we recognize the generating function for digraphs G with vertices \(\circ , \circ '\), \(1,\ldots , k\) that have edges \((\circ , \circ ')\) and \((\circ , j)\), \(j = 1, \ldots k\). The vertex \(\circ \) has color p, the vertex \(\circ '\) color \(p'\), and j the color \(x_j\).
We call such a digraph with the associated coloring a spike of type \((p,p')\) and the edge \((\circ , \circ ')\) the base edge of the spike. A spike with base edge \((\circ , \circ ')\) is identified with the enriched graph \({\bar{G}}\) that has the trivial partition consisting of a single block, \( P_{\circ } = \{\circ ', \varvec{x}_{[k]} \} \). See the left panel of Fig. 4 for an example of a spike.
Concatenating base edges of spikes in a circular fashion gives rise to an object consisting of a base cycle and additional incoming edges.
Definition 4.7
Let V be a finite nonempty set and \({\bar{f}} = (f,(P_v)_{v\in V}) \in \mathcal {M}[V]\). We call \({\bar{f}}\) a crown if there exists a finite set \(B\subset V\) (the base of the crown) such that:

\(f(V) = B\),

f restricted to B is a cycle,

for all \(r\in B\), \(P_r = \{f^{1}(\{r\})\}\), that is, the partition \(P_r\) consists of a single block.
Let \(\mathcal {K}[V]\) be the collection of crowns on V.
The right panel in Fig. 4 shows are crown. The simplest crown is a loop.
The next lemma expresses the relevant coefficient of the determinant as a sum over assemblies of crowns, with an additional factor \((1)^s\) where s is the number of crowns. The factor \((1)^s\) is a crucial ingredient to cancellations in the proof of Proposition 3.4.
Lemma 4.8
Let \(D_k\) be as in (4.15). Then,
where the sum is over set partitions \(\{V_1,\ldots ,V_s\}\) of [k].
Proof
Using the definition as formal power series (3.9), we see that the mth coefficient of the determinant is
The definition of the determinant gives
which we may also write as
with \(s(\sigma )\) the number of cycles in the cycle decomposition of the permutation \(\sigma \). Indeed, let \(\sigma =\sigma _1\cdots \sigma _s\) be the cycle decomposition into s cycles of respective lengths \(\ell (\sigma _i)\). Then, \(\sum _{i=1}^s \ell (\sigma _i)=m\) and
The sum in (4.22) is a sum over the following directed graphs. Each connected component is a cycle (or loop). Each edge \((p_1,p_2)\) contributes a factor \(M_{p_1,p_2}\) and the sign is \(1\) to the numbers of cycles. By the definition (A.2) of the product of power series, the nth coefficient of (4.22) at \((x_1,\ldots , x_n)\) is
The sum is over ordered set partitions \((J_1,\ldots , J_m)\) of [n] into m sets. Because of \(A_0=0\), only nonempty sets \(J_i \ne \varnothing \) contribute.
The expression (4.24) is a sum over permutations \(\sigma \in {\mathfrak {S}}_m\) and sequences \((G_1,\ldots , G_m)\) of spikes with vertex sets \(\{\circ _i \} \cup J_i\) with disjoint \(J_i\)’s and base edge \((\circ _{\sigma (i)}, \circ _{i})\). Such a sequence is naturally associated with the digraph G with vertices \(\{1,\ldots , n\} \cup \{ \circ _1 , \ldots , \circ _m\}\). Each connected component of G is a crown, the base of the crown consists only out of vertices from \(\{ \circ _1 , \ldots , \circ _m\}\) and all vertices not in the base are from \(\{1,\ldots , n\}\). The base of the crown corresponds to a cycle in \(\sigma \). The mapping \((\sigma , (G_1,\ldots , G_m))\mapsto G\) induces a onetoone correspondence between (i) pairs \((\sigma , (G_1,\ldots , G_m))\) and (ii) collections of crowns with the vertices in the bases of the crown are the vertices \(\{ \circ _1 , \ldots , \circ _m\}\) and all of them are in some base of a crown.
Hence, the nth coefficient of (4.20) is given by
where the primed sum is over enriched endofunctions on [k] whose connected components are crowns. The next step is to drop the distinction betweed \(\varvec{p}\) and \(\varvec{x}\), which gives rise to a binomial coefficient \(\left( {\begin{array}{c}n+m\\ m\end{array}}\right) \) which cancels with the factorial prefactors, that is 1/m! in (3.9), 1/n! from (3.5) and \(k!=(n+m)!\) from (4.15). The kth coefficient of \(\det (\mathrm {id}  M(z))\) is therefore equal to
\(\square \)
Proof of Proposition 3.4
Let \(E_k^n\) and \(D_k\) be the coefficients of the exponential and determinant as in (4.10) and (4.15). By (A.2), the power series in braces in Proposition 3.4 has as kth coefficient
In view of (A.4), we have to show, in order to prove Proposition 3.4, that
Note that \(\varvec{x}_I\) is replaced by \(\varvec{q}_I\) and k has to be equal to n. Hence, the list \(\varvec{q}_I\) is a part of \(\varvec{q}\).
By Lemmas 4.6 and 4.8 , the lefthand side of (4.27) is given by
In words, a sum over collections of crowns \({\bar{g}}_1,\ldots ,\bar{g}_r\) with disjoint supports \(V_1,\ldots , V_r\) and a forest \({\bar{f}}\) of cyclerooted trees with root cycle in \(I\subset [n]\setminus \cup _k V_k\) and vertexrooted trees with root vertices in \(\cup V_k\). Each such tuple can be mapped to a map \({\bar{F}}\in \mathcal {M}[n]\). The relation between weights is
The mapping \(({\bar{f}},{\bar{g}}_1,\ldots , {\bar{g}}_r)\mapsto {\bar{F}}\) is surjective but not injective: each \({\bar{F}}\in \mathcal {M}[n]\) is a forest of cyclerooted trees (no vertexrooted tree). Each such cyclerooted tree can decide whether (a) it belongs to the forest \({\bar{f}}\), or (b) it is split into a crown \({\bar{g}}\) and attached vertexrooted trees. Note that the set of splitting points is uniquely determined. The two choices for a given \({\bar{F}}\) come with opposite signs and sum to zero. \(\square \)
Forests with Several Sinks. Proof of Theorem 3.1
The proof of Theorem 3.1 builds on the cancellations from Proposition 3.4. The directed graphs in the proof of Proposition 3.4 consists of crowns and vertexrooted trees which can be combined in such a way that all connected components are cyclerooted trees. In contrast, for Theorem 3.1, the graphs consist of crowns and two types of vertexrooted trees. The first type is as in Proposition 3.4, the second one is related to the coefficients of the power series of \(\Phi \). As in Proposition 3.4, the cyclerooted trees cancel exactly with the first type of vertexrooted trees and only the second type of trees survives, which is a combinatorical justification that the inversion is related to vertexrooted trees only.
Let us introduce a shorthand for a forest of trees
where is the sum is over ordered partitions \((V_\ell )_{\ell \in L}\) of \([n]\setminus L\), with \(V_\ell = \varnothing \) allowed.
Lemma 4.9
Let \(n\in \mathbb {N}\) and \((q_1,\ldots , q_n) \in {\mathbb {X}}^n\). Fix \(L\subset [n]\) and consider sums over pairs \(I,J \subset [n]\) such that [n] is the disjoint union of L, I, and J. Then,
where \(I_1\), \(I_2\) are disjoint subsets of I such that \(I_1 \cup I_2 =I\) (the sets \(I_1, I_2\) could also be empty).
Proof
By Lemma 4.6 we have that
Define \(I_1\) as the union of all the vertices in I from connected components which are vertex rooted trees with root vertex in L. Then, \(I_2\) is the set of all vertices in I associated with root vertices in J. This correspondence is one to one. \(\square \)
Proof of Theorem 3.1
By the product formula (A.3), the nth coefficient of
at \((q_1,\ldots , q_n)\) is
where the sum is over ordered partitions L, I, J of [n] with empty sets allowed. By definition (3.4) of \(\zeta (\mathrm {d}q;\nu )\) and definition (A.10) of the composition, the nth coefficient of \(\Psi (\nu ) = \Phi (\zeta (\nu ))\) is
where the sum is over ordered partitions \((V_\ell )_{\ell \in L}\) of \([n]\setminus L\), with \(V_\ell =\varnothing \) allowed. So we have to show that the expressions (4.34) and (4.35) are equal. Indeed, by Lemma 4.9, we have that
Using that \(L, J, I_1, I_2\) are disjoint with union [n], we obtain that
Using the cancelations from Proposition 3.4 (most easily in the form (4.27)) we get that
where L and \(I_2\) are disjoint with union [n]. This is exactly (4.35). \(\square \)
Discussion
In this final section, we discuss connections to similar methods in different contexts. Some occurrences of trees in various areas of mathematics are listed in Table 1.
Gallavotti Trees
For simplicity, let us consider a gas consisting of classical particles interacting via a twobody potential V at inverse temperature \(\beta \). From basic statistical mechanics, we can write the density \(\rho \) as a function of the activity z as follows:
where \(b_n\) are the Mayer coefficients related to the pair potential V. We observe that one can invert the above expression following different strategies, we present a few. From (5.1), solving for z, we have:
By iterating over z and expanding the powers of the sums one obtains terms either with \(\rho (z)\) or with z in which case we keep expanding. It is easy to visualize this procedure; each iteration is a branching of a tree and each vertex of the children either has a \(\rho (z)\) (and we stop) or it has a z and we continue to the next generation. Hence, overall this expansion can be viewed as a power series ,in \(\rho (z)\) with the power representing the number of final points. This construction can be also found in in the Main Theorem 3 in [34] and in [6], see also the survey [35]. We note that this method of inverting power series has been already used in statistical mechanics, it is actually reminiscent of Gallavotti’s approach to express the Lindstedt perturbation series in the context of KAM theory [16].
We observe that in the above example trees are generated by iterating the mapping \(z\mapsto \rho \sum _{n\ge 2} n b_n z^n\) and they provide a power series representation of the fixed point solution of the mapping. We notice that other mappings can be suggested and, as it is usually the case, they correspond to more or less efficient methods. More precisely, some alternative mappings are
and
for some coefficients \(a_n\). In [23] we demonstrated, cf. Theorem 4.1, that (5.4) has a much better radius of convergence at least in some regimes.
Abdesselam’s Approach
In [3], A. Abdesselam presents an alternative proof of LagrangeGood multivariable inversion formula using a quantum field theory (QFT) model. Details are given in the companion papers [1] and [2]. This reveals an interesting connection between QFT calculations and Gessel’s combinatorial proof [17], and seems to show a similar kind of cancellations. One ingredient of Abdesselam’s proof is a representation of the onepoint correlation function of some complex bosonic field as a sum over trees, which connects to the representation of the inverse in terms of trees. Another ingredient is a graphical representation of a calculus of formal power series, coming with an algebraic formalization of Feynman diagrams [1].
Other Connections
Tree expansions is a favorite topic in several areas of mathematics. Without pretending of being exhaustive, we note Butcher series in computing higher order Runge–Kutta methods [9, 14] and the combinatorial structure in indexing Hepp sectors in renormalization and regularity structures [22]. A common feature is that they provide a power series representation of the solution of a fixed point problem and as such we believe expansion methods of this type can be widely used in applications. The techniques developed in this paper can be used to extend these expansion in an infinite dimensional context, for example in an inhomogeneous situation as in this paper.
Notes
The choice of letters \(\rho _i\) and \(z_i\) as well as the exponential form of the map are motivated by applications in statistical mechanics [26], where the \(z_i\)’s and \(\rho _i\)’s correspond to activity and density variables and the index i may refer to the type of a particle or a discrete set of locations on a lattice. The exponential form in Eq. (2.1) will be crucial in the following.
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Acknowledgements
S.J. thanks the GSSI, T.K. the University in L’Aquila, Italy, and D.T. thanks the LMU in Munich, Germany, for hospitality. S.J. was supported by the Munich Center for Quantum Science and Technology (MCQST). We thank Abdelmalek Abdesselam, Giovanni Gallavotti, and Alessandro Giuliani for fruitful discussions and Nils Berglund for pointing out the reference [22].
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Appendices
Appendix A: Formal Power Series
Here, we describe some operations on formal power series as introduced in Definition 2.1, e.g.,
where \(({\mathbb {X}},\mathcal {X})\) is a measurable space z is a measure on \(({\mathbb {X}},{\mathcal {X}})\), and \(K_0\in \mathbb {C}\) is a scalar, and \(K_n:{\mathbb {X}} ^n \rightarrow \mathbb {C}\) are measurable maps that are invariant under permutation of the arguments. Operations are defined purely in terms of the sequence of coefficients.
Product. Let K, G be formal power series, then KG is defined by
For a motivation of this definition, see e.g., [23, Appendix A]. The empty set \(J=\varnothing \) is explicitly allowed. As an operation on sequences of symmetric functions, this is exactly the convolution in [31, Chapter 4.4]. It is not difficult to check that the product is commutative and associative. Eq. (A.2) generalizes to products \(K^{(1)}\cdots K^{(r)}\) as
where the sum runs over ordered partitions \((V_1,\ldots , V_r)\) of [n] into r disjoint parts, with \(V_i=\varnothing \) explicitly allowed.
Variational derivative. For \(q\in {\mathbb {X}}\) and K a formal power series over \({\mathbb {X}}\), we define
For higherorder variational derivatives, see Definition 2.5.
Integrals. Measures \(K(\mathrm {d}q;z) = F(q;z) z(\mathrm {d}q)\). Let F(q; z) be a functionvalued power series. The power series
is defined as the power series with coefficients \(I_0:=0\) and
The definition is motivated by the following formal computation:
We also define the measurevalued formal power series \(K(\mathrm {d}q;z) = z(\mathrm {d}q) F(q;z)\) as the power series with coefficients \(K_0:=0\) and
Composition I and exponential series. Let \(F(t) = \sum _{n=0}^\infty f_n t^n/n!\) be a formal power series in a single variable t and K a formal power series on \(({\mathbb {X}},\mathcal X)\) with \(K_0=0\). The formal power series \(F\circ K\) on \({\mathbb {X}}\) is defined by \((F\circ K)_0:=f_0\) and for \(n\ge 1\),
with \({\mathcal {P}}_n\) the collection of set partitions of \(\{1,\ldots ,n\}\). Note that only because \(K_0=0\) the expression (A.8) is welldefined as a formal power series, because only in this case the sum is finite. An important special case is \(F(t) = \exp (t)\), for which Eq. (A.8) becomes
which is exactly the exponential on the algebra of symmetric functions from [31, Chapter 4.4]. For a motivation of this definition, see again, e.g., [23, Appendix A].
Composition II. In order to define the compositions in (3.3), we need a more general type of composition. Let G be a formal power series on \({\mathbb {X}}\) and F(q; z) a functionvalued power series
Let \(K(\mathrm {d}q;z)\) be the measurevalued formal power series \(K(\mathrm {d}q;z) = F(q;z)z(\mathrm {d}q)\) with coefficients (A.7). The composition \(H(z):= (G\circ K)(z)\) is defined as the power series with coefficients \(H_0=0\) and for \(n\ge 1\),
The summation is over partitions \((V_j)_{j\in J}\) of \([n]\setminus J\) with empty sets \(V_j = \varnothing \) allowed. Note that the sum is only over finitely many summands and hence welldefined as a formal power series. The definition is motivated by the following formal computation:
We group terms with the same sum \(n= m+ n_1+\cdots + n_m\), write
and note that the multinomial counts the number of ways to partition [n] into sets J and \((V_j)_{j\in J}\) in such a way that \(\#J = m\) and \(\#V_j = n_j\). Exploiting the symmetry of the integrands, we arrive at
with \(H_n\) defined in (A.10).
Composition of two measurevalued series. The composition (A.10) easily extends to the composition of a measurevalued formal power series \(\Gamma \) with a formal power series of the form \(K(\mathrm {d}q;z) = F(q;z) z(\mathrm {d}q)\), for which we have defined the coefficients in Eq. (A.7).
Then, \(H= \Gamma \circ K\) is the measurevalued series which has by Eq. (A.10) the coefficients for \(n\ge 1\) and \(B\subset {\mathbb {X}}\),
In treating the the compositions \(\rho \circ \zeta \) and \(\zeta \circ \rho \) in Eq. (3.3), we need to consider \(\Gamma \) which are of the form \( \Gamma (\mathrm {d}q;z) = G(q;z) z(\mathrm {d}q)\) for some function valued formal power series G, that is by Eq. (A.7) again
Then, also H is of the form \(H(\mathrm {d}q;z) = J(q;z) z(\mathrm {d}q)\), where J is a function valued formal power series and as above,
with \( H_0 =0\). Then, using Eq. (A.7), we can directly express
This also implies that
Restriction of power series Let \(\mathbb {Y} \subset \mathbb {X}\), then one can define a restriction of the formal power series to the set \(\mathbb {Y}\), denoted by \(K\!\!\upharpoonright _{ \mathbb {Y}}\), in the following way: restrict the coefficients of K to \(\mathbb {Y}\), that is, consider
One can consider \(K_n\upharpoonright _{ \mathbb {Y}} \) also as a function on \(\mathbb {X}^n\) by defining \(K_n\upharpoonright _{ \mathbb {Y}} =0\) outside of \(\mathbb {Y}^n\) and hence we may formally write
In case that the formal power series is an actual convergent power series, the above restriction corresponds to restricting the function \( z \mapsto K(z)\) to all measures which are zero outside of \(\mathbb {Y}\). In particular, when \(\mathbb {Y}\) contains only finitely many elements, then any measure zero outside of \(\mathbb {Y}\) is of the form \(z(\mathrm {d}x) = \sum _{y \in \mathbb {Y}} z_y\delta _{y}(\mathrm {d}x)\) for some \( z_y \in \mathbb {R}_+\) and hence, in this case, \(K\upharpoonright _{ \mathbb {Y}} \) can be seen as a function on \(\mathbb {C}^{\# \mathbb {Y}}\). The analogous construction works also for \(\mathbb {Y}\) with countable many elements, but not for uncountable many elements.
In order to compute the nth. coefficient of K evaluated at \(q_1, \ldots , q_n \in \mathbb {X}\), that is \(K_n(q_1,\ldots ,q_n)\), it is sufficient to consider \(K\upharpoonright _{ Q}\), where \(Q=\{q_i:\, i=1,\ldots ,n\}\) is the set of colors in \(q_1, \ldots , q_n\). Notice that the set of colors Q has cardinality smaller than n if colors are repeated in the vector, i.e., \(q_i = q_j\) for some \(i\ne j\). The relation between the coefficients is the following
where \(\varvec{z} := ( z_q)_{q\in Q}\), \(\varvec{n} := (n_q)_{q \in Q}\), and \(n_q:= \# \{ i \in \{ 1, \ldots , n \} \, : \, q_i =q \}\), that is the number of repetitions of the color \(q_i\).
This allows us to reduce the computation of the Fredholm determinant, which appears in Theorem 3.1, to the computation of usual determinants
Lemma A.1
Let \(n\in \mathbb {N}\) and \((q_1,\ldots ,q_n)\in {\mathbb {X}}^n\). Set \(Q=\{q_i:\, i=1,\ldots ,n\}\). Then the nth coefficient of the Fredholm determinant \(\det (\mathrm {Id} {\mathbb {K}}_z)\), evaluated at \((q_1,\ldots ,q_n)\), is equal to the nth coefficient at \((q_1,\ldots ,q_n)\) of the \((\#Q)\times (\#Q)\)matrix
Proof
We start with the more intuitive case of finite color set \({\mathbb {X}} = \{1,\ldots ,\ell \}\) with \(\ell \in \mathbb {N}\). In this case, the Fredholm determinant is just the determinant of an \(\ell \times \ell \) matrix,
The analogue of (2.17) reads
Suppose we want to know the coefficient of some monomial \(z_{1}^{n_1}\cdots z_\ell ^{n_\ell }\) in the determinant. Then, clearly the only relevant contributions are from summands \((x_1,\ldots , x_r)\) with every entry \(x_i\) contained in the support \(\mathrm {supp}\, \varvec{n} = \{x:\, n_x \ge 1\}\), which plays the role of Q. The series is actually a sum, as all terms for \(r > \#Q\) are zero. Noticing that
we conclude
Turning back to general sets \({\mathbb {X}}\), the result follows as the coefficient only depends on \(\det (\mathrm {Id}  {\mathbb {K}}_z) \upharpoonright _{Q}\) as discussed before the lemma. \(\square \)
Determinants for z either with density or which are generalized functions. In several applications, it is not natural to restrict z to points. In statistical mechanics for example, it is typical that \(\mathbb {X} \subset \mathbb {R}^d\) and z has a density with respect to the Lebesgue measure. For such measures z the restriction to Q is always zero. In quantum field theory, one considers z which are only generalized functions and the restriction to points has no sense at all. However, one can reinterpret the Fredholm determinant in another way and give an expression in terms of usual determinants. Assume that z either has a density with respect to a reference measure m or that z is a generalized function. The density we denote as well by z and the duality between test and generalized functions we formally write as an integral. The nth. coefficient of (2.17) depends only on
where \(z( q_1) \ldots z( q_n)\) are interpreted as the nfold tensor product of generalized functions. This shows that one can get the nth. coefficient by just computing the determinant of an \(n \times n\)matrix
The determinant is welldefined, because in all expressions the generalized functions z are evaluated at different points. The Dirac deltas cause no problem, because they only lead to dropping integrals. Indeed, one just obtain (A.22), which is welldefined for generalized functions z.
Appendix B: Combinatorial Species with Uncountable Color Space
Formal power series in finitely or countably many variables have a natural interpretation as exponential generating functions for labeled, colored combinatorial species, which helps prove identities of power series identities independently of any convergence considerations. This point of view is formalized with Joyal’s theory of combinatorial species [7, 24]. Power series in several variables correspond to colored combinatorial species, also called multisort species [7], with one variable \(z_k\) per color or sort. See the survey by Faris [13] for an account in the context of cluster expansions and [11] for applications to Feynman diagrams.
This appendix extends the theory of combinatorial species generalizes the theory of colored species to infinite, possibly uncountable color space \(C\). Such generalizations were in fact already proposed by Farris [12] as well as Méndez and Nava [28]; however, their concepts of generating function there are too restrictive for our purpose. Indeed their generating functions are sums of monomials \(\prod _{k\in C} z_k^{n_k}\) indexed by multiindices \(\varvec{n} = (n_k)_{k\in C}\) that have only finitely many nonzero entries \(n_k \ne 0\). Our generating functions instead are functions of measures \(z(\mathrm {d}x)\) on the color space \(C\). This is the kind of generating function that appears naturally in the statistical mechanics of inhomogeneous systems, where colors may correspond, for example, to positions \(x\in \mathbb {R}^d\) in space and the measure \(z(\mathrm {d}x)\) is a positiondependent activity [33].
Another difference with [28] is a more nuanced notion of family of power series. For finite color spaces and finitely many variables, it is natural to look at families \((F_k)_{k\in C}\) of generating functions, e.g., for rooted colored trees that have their root of color k. In our setup the relevant power series are either function or measurevalued. For example, rooted colored trees give rise to a family \((T_B^\bullet (z))_{B\subset C}\) indexed by sets \(B\subset C\) instead of elements \(k\in C\): to each set of colors \(B\subset C\) associate the generating function for trees whose root has color in B.
We proceed with a short selfcontained description of our formalism, which is similar to [28] but has a different notion of generating function. We start with the concrete example of graphs. Let \(C\) be a nonempty possibly infinite set, the set of colors. Let V be a finite set, e.g., \(V = \{1,\ldots ,n\}= [n]\) with \(n\in \mathbb {N}\); V is the set of labels. A \(C\)colored, labeled graph on V is a pair (G, c) consisting of (i) a graph G with vertex set V, (ii) a map \(c:V\rightarrow C\). Thus, every vertex of the graph is assigned both a label \(v\in V\) and a color \(c(v)\in C\). The pair (V, c) is a \(C\)colored set. We write \(\mathcal {G}(V,c)\) for the collection of colored labeled graphs on V with prescribed coloring c.
Admissible relabelings of vertices are formalized with bijections: Let \((W,{\tilde{c}})\) be another finite colored set and \(\varphi :V\rightarrow W\) a colorpreserving bijection, i.e., \({\tilde{c}}(\varphi (v)) = c(v)\) for all \(v\in V\). Relabeling the vertices v of a graph \(G\in \mathcal {G}(V,c)\) by \(w=\varphi (v)\) we obtain a graph \({\tilde{G}}\in \mathcal {G}(W,{\tilde{c}})\). In this way, \(\varphi \) induces a bijection between \(\mathcal {G}(V,c)\) and \(\mathcal {G}(W,{\tilde{c}})\). Choosing \(V=W=[n]\), we deduce that for every permutation \(\sigma \in \mathfrak {S}_n\), we have \(\#\mathcal {G}([n],c) =\#\mathcal {G}([n],c\circ \sigma )\). Put differently, \(\#\mathcal {G}([n],c)\) is a symmetric function of the variables \(c_i=c(i)\). The associated formal power series
is the (exponential) generating function of the species of labeled, \(C\)colored graphs.
In the following, we consider the set of colors C to be fixed once and for all.
Definition B.1
A colored set is a pair (V, c) consisting of a finite set V and a map \(c:V\rightarrow C\). A colorpreserving bijection \(\varphi :(V,c)\rightarrow (W,{\tilde{c}})\) is a bijection from V onto W such that \({\tilde{c}} = c\circ \varphi \).
The empty set \(V=\varnothing \) is considered a colored set. This is needed for the combinatorial counterpart of the variational derivative (see below) and allows for a conceptualization of pinned vertices that are not integrated over in generating functions. For example, we may be interested in the set of trees with vertex set \([n]\cup \{\star ,\circ \}\) where \(\star ,\circ \notin [n]\) are two distinct elements not in [n]. Then, \(n=0\) corresponds to trees with vertex set \(\{\star ,\circ \}\).
Sometimes we write colorings as vectors \((c_v)_{v\in V}\) and not as maps \(v\mapsto c(v)\).
Definition B.2
A labeled combinatorial species F with colors in C consists of two rules:

(1)
a rule \((V,c)\mapsto F(V,c)\) that assigns to every colored set a finite set (for \(V=\varnothing \) we write instead \(F(\varnothing )\)),

(2)
a rule that assigns to every colorpreserving bijection \(\varphi : (V,c)\rightarrow (W,{\tilde{c}})\) between colored sets a bijection \(\Phi :F(V,c)\rightarrow F(W,{\tilde{c}})\) (relabeling),
that satisfy the following: for all colorpreserving bijections \(\varphi _1:(V_1,c_1)\rightarrow (V_2,c_2)\), \(\varphi _2:(V_2,c_2)\rightarrow (V_3,c_3)\), the relabeling induced by \(\varphi _3=\varphi _2\circ \varphi _1\) is the composition of the relabelings, \(\Phi _3 =\Phi _2\circ \Phi _1\).
Put differently, a colored combinatorial species is a functor from the category of of colored sets (objects: finite colored sets, morphisms: colorpreserving bijections) to the category of finite sets. In concrete examples, the concept of relabeling and its functorial property are often so natural that the rule \(\varphi \mapsto \Phi \) is left implicit.
Definition B.3
Let F be a labeled colored combinatorial species. Suppose that for each colored map (V, c), we are given a weight function \(w_{(V,c)}:F(V,c)\rightarrow \mathbb {C}\) and that for all colorpreserving bijections \(\varphi :(V,c)\rightarrow (W,{\tilde{c}})\), \(w_{(W,{\tilde{c}})} \circ \Phi = w_{(V,{\tilde{c}})}\). The pair (F, w) of consisting of the species F and the family w of weights \(w_{(V,c)}\) is a weighted colored species.
By a slight abuse of notation we shall omit the indices and use the letter w both for the family of weight functions and for weight maps \(w:F(V,c)\rightarrow \mathbb {C}\).
Lemma B.4
Let (F, w) be a weighted colored species. Then for every \(n\ge 1\), every coloring \(c:[n]\rightarrow C\) and all \(\sigma \in \mathfrak {S}_n\),
Thus, \(\sum _{g\in G([n],c)} w(g)\) is a symmetric function of the color variables \(c(1),\ldots ,c(n)\) and we may apply the notion of formal power series. The elementary proof is left to the reader.
Definition B.5
The generating function of a weighted colored species (F, w) is the formal power series
Example B.6
(Colored singletons) Let \(B\subset C\) be a set of colors. The species of singletons with color in B is the species
The associated generating function is
compare Example 2.4.
Operations on formal power series correspond to operations on combinatorial species. We provide formulas for nonweighted species only, the generalization to weighted species is straightforward.
Cartesian product. Let F, G be two combinatorial species. We define a new species \(F\times G\) by
The generating function is \((F\times G)(z) = F(z) G(z)\), compare Eq. (A.2). Hence, an \(F\times G\)structure on V is a pair (f, g) consisting of an Fstructure on \(V_1\) and a Gstructure on \(V_2\), with \(V_1,V_2\) a partition of V into possibly empty sets.
Derivatives. Let \((V,c)\mapsto G(V,c)\) be some colored combinatorial species. Suppose that for each finite set V there is a designated element \(\circ = \circ _V\) that is not in V, see Definition 5, Remark 6, and Exercise 16 in [7, Chapter 1.4]. Given \(q\in C\), we extend a coloring \(c:V\rightarrow C\) to a coloring \(c_q^\circ : V\cup \{\circ \} \rightarrow C\) by setting
Then we define a family \((F_q)_{q\in C}\) of colored species by
The generating function is
in which we recognize the variational derivative
see Eq. (A.4). We may think of objects in \(G_q^\circ \) as objects of G rooted in \(\circ \). For example, when \(G= T\) is the species of nonrooted trees, then \(T_q^\circ \) can be identified with the species of trees rooted in the nonlabeled vertex (ghost) \(\circ \) with prescribed root color q.
Pointing. Let G be a species and (V, c) a finite colored set. For \(B\subset C\), define
For example, when \(G=T\) is the species of the nonrooted trees, \(T_B^\bullet \) corresponds to rooted trees with labeled root and root color in B. We have
Comparison with Eq. (A.7) yields
Compare Lemma 4.4 and the paragraph preceding the lemma.
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Jansen, S., Kuna, T. & Tsagkarogiannis, D. Lagrange Inversion and Combinatorial Species with Uncountable Color Palette. Ann. Henri Poincaré 22, 1499–1534 (2021). https://doi.org/10.1007/s00023020010130
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DOI: https://doi.org/10.1007/s00023020010130
Mathematics Subject Classification
 32A05
 47J07
 05A15
 82B21
 81T15