The tensor Harish-Chandra-Itzykson-Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers

We study a generalization of the Harish-Chandra - Itzykson - Zuber integral to tensors and its expansion over trace-invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which count certain families of constellations. We find an expression of these numbers in terms of monotone simple Hurwitz numbers, thereby also providing expressions for monotone double Hurwitz numbers of arbitrary genus in terms of the single ones. We give an interpretation of the different combinatorial quantities at play in terms of enumeration of nodal surfaces. In particular, our generalization of Hurwitz numbers is shown to enumerate certain isomorphism classes of branched coverings of a bouquet of $D$ 2-spheres that touch at one common non-branch node.


Introduction
The problem.Our goal is to explore the following generalization of the HCIZ integral [31,32]: (1.1) We will be particularly interested in the expansion of its logarithm, C D,N pt, A, Bq " log I D,N pt, A, Bq, as a power series in t and a Laurent series in N and in its behavior at large N .For D " 1 we take U a unitary N ˆN matrix U P U pN q (U ˚" U ´1, where star denotes the adjoint), A and B self-adjoint N ˆN matrices and E U the expectation with respect to the Haar measure; that is for D " 1 (1.1) is the usual HCIZ integral [31,32], here denoted by I 1,N pt, A, Bq.
In this paper we are interested in the setup where U is a tensor product of D ě 2 unitary matrices U " U p1q b . . .b U pDq with U pcq P U pN q, E U is the expectation with respect to the tensor product of D Haar measures and A, B are self-adjoint N D ˆN D matrices called the external tensors.In this case we will call (1.1) the tensor HCIZ integral.This is also written as: I D,N pt, A, Bq " e C D,N pt,A,Bq " ż rdU s e tTrpAU BU ˚q . (1.2) Let us note that if the tensors A, B belong to the Lie algebra of U pN q bD and are generic (i.e. in the interior of a Weyl chamber), the integral I D,N pt, A, Bq admits an exact determinantal formula as per Harish-Chandra's general results.In the particular case of D " 1, this statement is important because it allows handling all multiplicity-free self-adjoint matrices A, B. For D ě 2 however, the Lie algebra is smaller and the problem we are considering is much more general.
Motivations.The Kontsevich integral [33] for a N ˆN self-adjoint matrix X: with B a fixed N ˆN matrix is the prototype of a non-invariant probability distribution for a random matrix.This integral plays a crucial role in two dimensional quantum gravity [16].The Grosse-Wulkenhaar model [23] is obtained by replacing the cubic potential with a quartic one and for a specific choice of B this model is a φ 4 quantum field theory on the non-commutative Moyal space [42] expressed in a matrix base [17,19].Such models can be generalized [3,10,20] to rank D complex tensors T (with components denoted by T i 1 ...i D ) transforming in the Dfundamental representation of the unitary group (T 1 i 1 ...i D " One is then interested in partition functions of the type: Z " ż dT d T e ´T B T `V pT, T q , T B T " where T denotes the dual of T (with complex conjugated entries T i 1 ...i D and transforming in the conjugated representation) and B some N D ˆN D matrix.The crucial point is that the perturbation V pT, T q is taken to be invariant under the action of U pN q bD .A striking feature of both the Kontsevich integral and its generalizations involving random tensors is that one considers a non-invariant quadratic part and an invariant interaction.In order to study the interplay between these two, one can average over the unitary group: Z " ż dT d T e V pT, T q ż rdU s e ´T U BU ‹ T .
The integral over U is then just a particular case of (1.2) for A the tensor product of T and its dual.
More generally, the HCIZ integral is extensively used in D " 1 in random matrix models with non-invariant probability distributions, such as the two-matrix models [32,38,22], and matrix models with an external source [6,7,44,45,8,4], to cite just a few references.It is also central in studying the law of Gaussian Wishart matrices and non-centered Gaussian Wigner matrices [30].The study of the tensor HCIZ integral (1.1) is justified by the generalization of these problems to random tensors.For example, it is natural in Quantum Information to study the sum of independent random tensors, so as soon as they have a U pN q bD -conjugation invariance, we expect that our results are a necessary preliminary towards to study of such asymptotic models.
Another application of the HCIZ integral is as a generating function for the monotone double Hurwitz numbers [24,25].Hurwitz numbers count n-sheet coverings of the Riemann sphere by a Riemann surface of a certain fixed genus, where one branch point, for instance 0, is allowed arbitrary but fixed ramification, and all the other branch points are only allowed simple ramifications.Double Hurwitz numbers are such that not one but two points on the sphere, say 0 and 8, are allowed non-simple ramifications [24,25,26,27].Monotone single and double Hurwitz numbers are such that only a subset of the possible coverings are allowed.These numbers appear in D " 1 as coefficients when expanding the HCIZ integral on the trace invariants of the two external matrices [24,25].
It is this last aspect on which the emphasis is put in the present paper: we expand the logarithm of the tensor HCIZ integral on the trace invariants of the two external tensors and study the 1{N expansion of the coefficients.This provides higher order generalizations of monotone double Hurwitz numbers.We provide a detailed study of the geometrical interpretation of these numbers and their relation to enumerations of branched coverings of a bouquet of D spheres.
Final comments.Before proceeding let us comment some more on our model: Different N 's.The generalization to the case of different dimensions U pcq P U pN c q is straightforward.
The D=1 case.For D " 1 we get the HCIZ integral which is a Fourier transform of the U pN qinvariant probability measure concentrated on the orbit of the matrix B. In general, the same holds true, but for the smaller symmetry group U pN q bD instead of U pN D q.
Variants.Other models might be relevant, such as: where A, B are tensors.For D " 1 this model boils down to the Brézin-Gross-Witten (BGW) integral t Ñ log E U " exp `tTrpAU `U ˚B˚q ˘‰ .
which was largely studied in the literature.For an optimal analytic result both for HCIZ and BGW in the D " 1 setup, we refer to [40].It turns out that the combinatorics are a bit different and slightly more involved, so we will move to this model in subsequent work.
Other groups.Similar results can be derived for the orthogonal or symplectic group.The initial theory does not change substantially, but the graphical interpretation does.We also keep this for future work.
Plan of the paper.The notations and prerequisite on Weingarten calculus, constellations and cumulants are gathered in Sec. 2, where the reader will also find, in Prop.2.5, an expression for the moments of the tensor HCIZ integral which follows directly from the definitions.
The study of the cumulants of the tensor HCIZ integral is more involved.They write in terms of a cumulant Weingarten functions, defined in Sec.3.1 and expressed in Sec.3.2 as series in powers of 1{N whose coefficients p C enumerate certain transitive factorizations of D-uplets of permutations.The rest of the paper is dedicated to the study and interpretation of the coefficients p C .Sec. 4 contains our main theorem, Thm.4.1.This theorem expresses the coefficients p C as sums over partitions satisfying certain conditions.In this form we are able to compute p C at leading order in 1{N , that is we identify the smallest exponent of 1{N with non vanishing contribution to p C and compute this contribution.
In D " 1, the coefficients p C are related to monotone double Hurwitz numbers, as detailed in Sec.4.3.These numbers are known to count certain isomorphism classes of connected branched coverings of the Riemann sphere.For D ą 1, the coefficients p C lead to a generalization of monotone double Hurwitz numbers, and one may wonder whether these numbers have a natural interpretation as enumerating certain branched coverings.This question is addressed in Sec. 5.After introducing nodal surfaces in Sec.5.1 we shown in Sec.5.2 that the generalized Hurwitz numbers of Sec.4.3 enumerates certain connected branched coverings of D 2-spheres that "touch" at one common node (a bouquet of D 2-spheres).This provides (see Sec. 5.3) a geometric interpretation for the combinatorial formulas of Thm.4.1 and recasts the sums over partitions defining p C as a sum over certain nodal surfaces whose nodes are weighted with monotone single Hurwitz numbers.

Notations
Indices ranging from 1 to N will be denoted by a 1 , a 2 , b 1 , b 2 and so on.Let S n be the group of permutations of n elements and S n the set of permutations different from the identity, S n " S n ztidu.For σ P S n , #pσq denotes the number of disjoint cycles of σ and σ the number of transpositions of σ (i.e. the minimal number of transpositions required to obtain σ) 1 .These quantities satisfy the identity: #pσq ` σ " n . (2.1) We denote by pρ 1 , . . ., ρ k q " pρ i q 1ďiďk , ρ i P S n or sometimes ρ an ordered sequence of k permutations, that is a constellation (see Sec. 2.5).
In this paper, we will deal with indices, permutations, and sequences of permutations bearing a color c P t1, . . ., Du.The color is indicated in superscript or subscript: a c 1 are indices, σ c are permutations, and so on.D-uplets will be written in bold, for instance σ " pσ 1 , . . ., σ D q, σ c P S n is a D-uple of permutations (S n denotes the set of pn!q D such D-uplets), and ρ is a Duple of constellations.For σ, τ P S n , we denote by ν " στ ´1 the D-uple of permutations ν " pσ We denote by π, π 1 , π 1 , π 2 and so on partitions of the set t1, . . ., nu and Ppnq the set of all such partitions.The notation |π| is used for the number of blocks of π, while B P π denotes the blocks, and |B| the cardinal of the block B. ď signifies the refinement partial order: π 1 ď π if all the blocks of π 1 are subsets of the blocks of π.Furthermore, _ denotes the joining of partitions: π _ π 1 is the finest partition which is coarser than both π and π 1 .Let 1 n be the one-block partition of t1, . . ., nu.
The partition induced by the transitivity classes of the permutation ν (i.e. the disjoint cycles of ν) is denoted by Πpνq, hence |Πpνq| " #pνq.d p pνq denotes the number of cycles of ν with p elements (d 1 pνq is the number of fixed points of ν) and we have: Note that if π ě Πpνq, then ν stabilizes the blocks of π, that is νpBq " B for all B P π.
Finally, Πpσ, τ q denotes the partition induced by the transitivity classes of the group generated by tσ c , τ c | c P t1, . . ., Duu, that is, Πpσ, τ q " Ž D c"1 `Πpσ c q _ Πpτ c q ˘and |Πpσ, τ q| is its number of blocks.Note that all the permutations in σ, τ stabilize the blocks of some partition π if and only if Πpσ, τ q ď π.

Trace invariants
We are interested in the invariants that can be built starting from a N D ˆN D matrix A. We define the trace invariant associated to σ P S n as: These quantities are obviously invariant under conjugation by U pN q bD , that is A Ñ U AU ‹ with U " U p1q b . . .b U pDq , U pcq P U pN q.For example: -for D " 1 any Tr σ pAq is a product of traces of powers of A, and the powers are the lengths of the cycles of σ: For n " 5, D " 1 and σ " p123qp45q we get Tr σ pAq " TrpA 2 q TrpA 3 q .
-if all the σ's are equal, σ c " σ, then Tr σ pAq is again a product of traces of powers of A, but this time the traces are over indices of size N D .Taking as before σ " p123qp45q (n " 5 and D arbitrary) we get Tr pσ,...,σq pAq " TrpA 2 q TrpA 3 q.
-we finish by an example with different σ's.For n " 2, D " 2, σ 1 " p12q, and σ 2 " p1qp2q: where Tr c denotes the partial trace on the index of color c.

Weingarten calculus
Weingarten calculus [43] allows one to integrate any polynomial function on the unitary group.There exists a function W pN q : S n Ñ R such that, denoting dU the Haar measure on U pN q, we have [13]: 3) The function W pN q is uniquely defined if and only if n ď N , and it follows from obvious commutativity relations that W pN q pστ ´1q depends only on the conjugacy class of στ ´1, that is W pN q is a central function on the symmetric group S n .The functions W pN q are called Weingarten functions.
The 1{N expansion of the Weingarten functions.We start with a theorem that characterizes and defines the Weingarten functions.Multiplying (2.3) by ś n s"1 δ as,bs and summing the repeated indices, we get: Theorem 2.1 (Collins-Śniady [14]).The Weingarten function ν Ñ W pN q pνq and the function ν Ñ Φpνq " N #pνq " N n δ id;ν `N n ř ρ‰id N ´ ρ δ ρ;ν are pseudo-inverses for the convolution.In particular, one has for N ě n: This theorem can be used to compute the Weingarten functions: with the convention that empty products are 1 and empty sums are 0. The case k " 0 writes ν as an empty product in S n , hence forces ν " id and the empty sum The coefficient of N ´n´l in the 1{N expansion of W pN q pνq is identified as [13]: W pN q pνq " N ´n ÿ lě0 p´1q l ppν; lq N ´l , p´1q l ppν; lq " ÿ kě0 p´1q k mpν; l, kq, where: mpν; l, kq " Card " pρ i q 1ďiďk ˇˇˇρ i P S n , with ρ 1 ¨¨¨ρ k " ν and and for l " 0 or 1, we have respectively mpν; 0, kq " δ ν,id δ k,0 and mpν, 1, kq " δ k,1 δ ν ,1 , and for k " 0 or 1, we have respectively mpν; l, 0q " δ ν,id δ l,0 and mpν, l, 1q " δ l, ν p1 ´δν,id q.We conclude that ppν; 0q " δ ν,id and ppν, 1q " δ ν ,1 .This expression for the coefficient at order N ´n´l as an alternating sum does not render explicit its sign.Another expression [15] (see also [37]) solves this issue.Definition 2.2.Let ppqq be the elementary transposition of p and q (that is we use a cycle notation, but we omit the cycles with 1 element).An ordered l-tuple of transpositions µ 1 " pp 1 q 1 q, . . ., µ l " pp l q l q is said to have weakly monotone maxima if p k ă q k for each k P t1, . . ., lu and q k ď q k`1 for each k P t1, . . ., l ´1u.15,37]).We denote by P pν; lq the set of solutions of ν " µ 1 . . .µ l with µ 1 , . . ., µ l elementary transpositions with weakly monotone maxima.Then ppν; lq " |P pν; lq|.
Asymptotics of the Weingarten functions.Classical theorems in combinatorics allow one to obtain the asymptotics of the Weingarten functions (Theorem 2.15 point piiq in [13]).
Corollary 2.4.For ν P S n , we have the asymptotic expansion: where Mpνq is Biane-Speicher's Möebius function on the lattice of non-crossing partitions ( [39], Lecture 10) which is a central function which can be written in terms of the Catalan numbers:

Moments of the tensor HCIZ integral
The moments of the tensor HCIZ integral (1.1) write in terms of the Weingarten functions.
Proof.The proof is straightforward.Starting from: and using D times the Weingarten formula (2.3), the expectation amounts to: where we recognize the definition of the trace invariants (Sec.2.2).Observe that the first and the second index of the U 's in (2.3) play slightly different roles, leading to the fact that the permutations for the invariant of B are inverted.
From Corollary 2.4, we obtain the asymptotic expression of the moments:

Constellations
We now review some results on the enumeration of constellations.Constellations are central to the combinatorial interpretation of our main results.
Definition and graphical representation.Intuitively, a combinatorial map (fatgraph, or ribbon graph in the physics literature) is a graph embedded in a closed surface2 in which each edge is subdivided into two half -edges.The map is bipartite if its vertices have one of two flavors (say 1 and 2) and every edge connects two vertices of different flavors.Formally a bipartite combinatorial map, or a 2-constellation, is an ordered pair of permutations ρ " pρ 1 , ρ 2 q, ρ 1 , ρ 2 P S n .It is represented canonically as an embedded graph as follows: • we let the flavor i " 1, 2. For each cycle of ρ i we draw a vertex embedded in the plane (a disk).For each s P t1, . . ., nu we attach a half-edge, i.e. an outgoing segment to one of the vertices, labeled xsy i .Every s belongs to a cycle of ρ i and we draw the half-edges xsy i , xρ i psqy i , xρ i pρ i psqqy i and so on ordered cyclically counterclockwise around the vertex corresponding to this cycle.
• for every s P t1, . . ., nu we join the two half-edges xsy 1 and xsy 2 into an edge labeled s.
The permutations ρ 1 and ρ 2 encode the "successor" half-edge: xρ i psqy i is the first half-edge encountered after xsy i when turning counterclockwise around the vertex of flavor i to which xsy i belongs.The permutation ρ 1 ρ 2 maps the edge s onto the edge ρ 1 pρ 2 psqq obtained by first stepping from s to ρ 2 psq, the successor of s on the vertex of flavor 2 to which s is hooked, and then stepping from ρ 2 psq to ρ 1 pρ 2 psqq, the successor of ρ 2 psq on the vertex of flavor 1 to which ρ 2 psq is hooked.The cycles of ρ 1 ρ 2 are the faces of the map.This can be generalized to k ě 2 flavors.A labeled k-constellation is an ordered k-uple of permutations ρ " pρ 1 , . . ., ρ k q, ρ 1 , . . ., ρ k P S n .The construction below is exemplified in Fig. 1, which we will be using extensively: (3) (24) Figure 1: A constellation with k " 3 flavors and n " 5 with ρ 1 " p12qp354q, ρ 2 " p1qp253qp4q and ρ 3 " p134qp2qp5q.Its faces are the cycles of ρ 1 ρ 2 ρ 3 " p1qp24qp3qp5q, corresponding to three hexagons and one dodecagon.The labels in the figure are respectively the flavors of the vertices (no parenthesis) the labels of some of the half edges (angled brackets), the labels of the white vertices (square brackets), and the cycles of ρ 1 ρ 2 ρ 3 corresponding to the faces (parenthesis).
• To each permutation ρ i , i P t1, . . ., ku we associate a set of embedded vertices of flavor i corresponding to its cycles.In Fig. 1 the flavored vertices are represented as blue (for flavor 1), red (for flavor 2) and yellow (for flavor 3).As ρ 1 " p12qp354q there are two blue vertices in the figure, one bi-valent corresponding to the cycle p12q and one tri-valent corresponding to the cycle p354q.
The vertices of flavor i have a total of n outgoing, cyclically ordered counterclockwise, halfedges xsy i for s P t1, . . ., nu.For instance, in Fig. 1, the three half-edges incident to the tri-valent blue vertex are labeled x3y 1 , x5y 1 and x4y 1 .
• To each s P t1, . . ., nu we associate an embedded white vertex rss. 3 We connect the half edges xsy i for all the flavors i to the vertex rss via edges such that the flavors are encountered in the order 1, . . .k when turning around the vertex rss clockwise, see Fig. 1.We label the edges by their end vertices as prss, iq or pi, rssq.
• The faces of the constellation are the disjoint cycles of the product ρ 1 ¨¨¨ρ k .A cycle of length p corresponds to a face with 2pk corners (bounded by 2pk edges) that are alternatively white vertices rss and flavored vertices corresponding to ρ i .The flavored vertices are encountered cyclically in the order k, k ´1, . . .2, 1 when going around the face while keeping the boundary edges to the left.
• The connected components of the resulting graph correspond to the transitivity classes of the group generated by tρ 1 , . . ., ρ k u.Indeed for s, s 1 P t1, . . ., nu such that ρ i 1 ¨¨¨ρ i l psq " s 1 , the vertices rss and rs 1 s are connected in the constellation by the path: prss, i l qpi l , rρ i l psqsq . . .prρ i 2 ¨¨¨ρ i l psqsi 1 qpi 1 rs 1 sq .
The partition Πpρ 1 , . . ., ρ k q " Ž k i"1 Πpρ i q " Πpρq is reconstructed by collecting all the white vertices rss belonging to the same connected component of the constellation into a block.The number of connected components of a constellation is |Πpρq|.The constellation is said to be connected if |Πpρq| " 1, that is the group generated by tρ 1 , . . ., ρ k u acts transitively on t1, . . ., nu.
For k " 2 the white vertices rss have valency 2 and therefore can be viewed as decorations (bearing labels) on edges, and we recover the bipartite maps described at the beginning of the section.
Euler characteristic.A k-constellation ρ has ř i #pρ i q `n vertices, kn edges, #pρ 1 ¨¨¨ρ k q faces and |Πpρq| connected components.Being a combinatorial map, it has a non-negative genus, denoted by gpρq ě 0, and Euler characteristic: #pρ i q ´npk ´1q `#pρ 1 ¨¨¨ρ k q " 2|Πpρq| ´2gpρq . (2.10) A constellation (seen as a combinatorial map) is planar, gpρq " 0, if and only if it can be drawn on the 2-sphere without edge-crossings such that each region of the complement of the graph on the sphere is homeomorphic to a disc.Stated in terms of the length of the permutations, (2.10) becomes: Among the connected constellations, the planar ones are such that Enumeration of planar constellations.The main result we will need is due to [5] and concerns the enumeration of planar constellations.We fix ν P S n .For k ě 2, the number of connected planar k-constellations pρ 1 , . . ., ρ k q in S n with faces corresponding to disjoint cycles of ρ 1 ¨¨¨ρ k " ν is: For the boundary values, we get: This can be adapted for constellations pρ 1 , . . ., ρ k q satisfying the same assumptions, but for which none of the permutations ρ i involved is the identity [5] (the constellations are said to be proper ), whose number is given by: γpν; kq " k ÿ j"0 ˆk j ˙γpν; jqp´1q k`j , ( and γpν; 0q " δ ν;id .One can furthermore compute [13] the following alternating sum: with Mpνq the Möebius function on non-crossing partitions (2.7).Note that this is a class function.
More generally, we denote by γl pν; kq and γ l pν; kq the numbers of generic (i.e.not necessarily proper) and respectively proper connected k-constellations with faces corresponding to the disjoint cycles of ρ 1 ¨¨¨ρ k " ν and with ř k i"1 ρ i " l (hence genus 2gpρq " l `2 ´2n ` ν ): where the last equality follows by inverting the relation γl pν; kq " ř k j"0 `k j ˘γl pν; jq.Finally, we denote by γ l pνq the alternating sum of the numbers of connected proper constellations with faces corresponding to the disjoint cycles of ρ 1 ¨¨¨ρ k " ν and ř k i"1 ρ i " l: Eq. (2.12), (2.14), and (2.15) correspond to the minimal possible value l " 2n ´2 ´ ν .

Cumulants
For X some random variable, the cumulant CpX n q, also sometimes called connected correlation, is defined by: For instance, the second cumulant CpX 2 q " EpX 2 q ´EpXq 2 is the variance of the probability distribution of X.The cumulants write in terms of the moments of the distribution and vice versa.In order to write down the relation between the two in a convenient form, we introduce some notation.
It is convenient to distinguish between the different factors X in the monomial X n .We do this by introducing a fictitious label i " 1, . . .n and writing X n " X 1 ¨¨¨X n where X i " X for all i.Then the n'th cumulant can be written as CpX n q " Cp ś n i"1 X i q.For any partition π P Ppnq with blocks B P π, we define C π " ś BPπ C p ś iPB X i q.We are now in the position to write the expectations in term of the cumulants: (2.17) The equation (2.17) can be inverted through the Möebius inversion formula to yield the cumulants in term of the expectations.Defining E π " ś BPπ E p ś iPB X i q, we have: where λ π 1 ,π is the Möebius function for the lattice of partitions [41] 4 : where π 1 |B is the restriction of the partition π 1 to the block B P π.This restriction is well defined because π 1 ď π.In particular, recalling that 1 n denotes the one block partition, we have:

Cumulants of the tensor HCIZ integral
The study of the cumulants of the tensor HCIZ integral is the core of this paper.They expand in terms of trace invariants of A and B times cumulant Weingarten functions defined in Sec.3.1.
An expression of the latter as a series in 1{N is derived in Sec.3.2.The coefficients of this expansion are shown to count certain transitive factorizations of D-uplets of permutations.

The cumulant Weingarten functions
We denote χp¨q the indicator function which is one if the condition ¨is true and zero otherwise.
Definition 3.1 (The cumulant Weingarten functions).For any partition π, let W pN q π rσ, τ s be: -zero if at least one of the permutations involved in σ or τ does not stabilize the blocks of π, that is W pN q π rσ, τ s is zero unless Πpσ, τ q ď π.
-the product over the blocks of π of Weingarten functions involving permutations restricted to these blocks if all the permutations in σ, τ stabilize the blocks of π.
Denoting σ c|B the restriction of σ c to the block B P π (which is well-defined whenever σ c stabilizes the blocks of π), we have: The cumulant Weingarten function W pN q C rσ, τ s is: where λ π " λ π,1n is the Möebius function with the second argument set to the one-block partition.
Observe that, due to the indicator function, both W pN q π rσ, τ s and W pN q C rσ, τ s depend on σ and τ and not only on the product στ ´1.The cumulant Weingarten functions arise naturally in the expansion of the cumulants of the tensor HCIZ integral over trace invariants.Proposition 3.2.The cumulants of the tensor HCIZ integral (1.1) are: where W pN q C rσ, τ s is the cumulant Weingarten functions, uniquely defined for N ě n.Proof.To any partition π, we associate the expectation: where the second equality follows from Prop.2.5.It then follows from (2.19) that: which leads to Eq. (3.2) using the definition (3.1).

Exact expression of the cumulant Weingarten functions
Theorem 3.3.The cumulant Weingarten functions are: where: and m C pσ, τ ; l, kq is the number of D-uplets of constellations pρ c ic q 1ďicďkc , c P t1, . . ., Du, with the following properties: • all the permutations ρ c ic are different from the identity permutation, • for all c P t1, . . ., Du, • the collection of all σ c , τ c , pρ c ic q 1ďicďkc ( 1ďcďD , acts transitively on t1, . . ., nu.This expansion is convergent for N ě n.
Proof.The functions W pN q π rσ, τ s in Def.3.1 are non-trivial only if σ and τ stabilize the blocks of π.We denote by ν " στ ´1 and ν c|B the restriction of ν c to the block B. (2.4) leads to: where we have exchanged the sums and the products.Note that if k B c " 0 for some c and B, then there are no permutations ρ c,B i B c and the rightmost sum becomes δ ν c|B ;id .
The permutations ρ c,B i B c can be trivially lifted to permutations on t1, . . ., nu by supplementing them with the identity on the complement of B. We denote the set of all the (lifted) permutations ρ by: and Πp ρq the partition induced by the transitivity classes of the group generated by all the permutations in ρ.As ρ c,B i B c acts non-trivially only on the block B P π, it follows that all the permutations in ρ stabilize the partition π, hence Πp ρq ď π.Now comes the subtle point.We would like to rewrite W pN q π rσ, τ s via a moment-cumulant formula such as (2.18), that is as a sum over π 1 ď π of "cumulants".The obvious idea to reorganize the sum by the partition Πp ρq of a summand which in turn sums all the ρs with the same Πp ρq fails due to the global factor χ `Πpσ, τ q ď π ˘.The second idea works: we reorganize the sum by the partition Πpσ, τ q _ Πp ρq ě Πpσ, τ q, that is we note that: with the cumulant: This expression is inverted using (2.18) to yield: Choosing π " 1 n , we recover the right hand side of (3.1), thus C,1n rσ, τ s, i.e.: and we recognize the coefficient of N ´nD´l in this expansion to be the alternating sum defining m C pσ, τ ; l, kq.
One drawback of Eq. (3.5) is that analytic bounds are difficult to obtain because the sum is signed.On the other hand, it renders obvious the invariance by relabeling of t1, . . ., nu.
There is a non-signed version in terms of a generalization of monotone double Hurwitz numbers, which we describe now (see also Sec. 4.3).
In particular p C rσ, τ ; ls is a non-negative integer.
Proof.Note that where P pν, lq denotes the set of transpositions with weakly monotone maxima which factorize ν (Def.2.2).From this point, the proof is mutatis mutandis the same as that of Thm.3.3.
4 Asymptotics of the cumulant Weingarten functions.
In many applications, such as random tensor models, one is interested in the first place in computing the large-N contribution to the logarithm C D,N pt, A, Bq of the tensor HCIZ integral (1.1).For some given σ, τ P S n , one thus needs to identify the smallest integer l such that p C rσ, τ ; ls does not vanish and, if possible, to obtain an explicit expression for the corresponding p C .We provide this in Thm.4.1.A general combinatorial formula, (4.6), is furthermore derived for p C rσ, τ ; ls for any l.To our knowledge this expression for the sub-leading contributions to the cumulant Weingarten functions is new also in D " 1.

Main result
The large N behavior of the cumulant Weingarten functions is captured by the following theorem.
Theorem 4.1.For any l, the coefficient p C rσ, τ ; ls is given by: with ν c " σ c τ ´1 c and γ l pνq defined in Sec.2.5.The smallest value of l such that p C rσ, τ ; ls does not vanish is: In order to simplify the notation we sometimes denote " pσ, τ q.The cumulant Weingarten functions thus have the asymptotic expression: where the leading order coefficient is: Note that ν c|Bc is well-defined as π c ě Πpν c q.In detail: with the non-crossing Möebius function M defined in (2.7).
The sum over partitions appears rather complicated, however it has a simple graphical interpretation in terms of sums of trees.This graphical interpretation was developed in [46] in D " 1 and for l " pσ, τ q and is generalized in this paper to larger l and larger D in Sec. 5 (more precisely Sec.5.3.2).Corollary 3.4 implies that if |Πpσ, τ q| " 1 then p´1q p C rσ, τ , s " ś D c"1 Mpν c q.We have chosen to factor the Möebius functions to render this explicit.This can be obtained directly from (4.4): as π c ě Πpν c q, we have |Πpν c q| ě |π c | and, from the condition in the sum, |Πpν c q| " |π c | so that only π c " Πpν c q contributes, and ν c|Bc is the restriction of ν c to one of its cycles: The leading contribution at large N to the cumulant is: As a function of the scaling behavior of the trace invariants Tr σ pAq and Tr τ ´1 pBq with N , the sum in (4.7) is dominated by a subset of the terms.For instance if Tr σ pAq " Tr τ ´1 pBq " Op1q in the limit of large N , then the term with σ " τ , }Πpσq} " 1 will dominate.If Tr σ pAq " 1 but Tr τ ´1 pBq " N ř c #pτcq , more terms dominate at large N .A detailed study of the possible behaviors of the cumulant, relevant for different applications to physics will be conducted in future work.

Proof of Theorem 4.1
The proof of Theorem 4.1 is divided into four parts: -Derivation of (4.1).Although lengthy, this part is straightforward: we compute the sum over k in Theorem 3.3.We obtain (4.1)where the right hand side is an alternating sum (with constraints) over constellations.
-Reinterpretation of Πpσ, τ q _ π 1 _ . . ._ π D " 1 n .We show that the condition: which constrains the sum over tπ c u c in (4.1) is equivalent to requiring that a certain abstract graph, aptly denoted G " Πpσ, τ q, tπ c u c ; Πpν c q ‰ , is connected.
-The lower bound (4.2) on l.We show that the leading order at large N (minimal l) in (4.1) fulfills two conditions: the abstract graph G " Πpσ, τ q, tπ c u c ; Πpν c q ‰ has minimal number of edges.As it is connected, this means it is a tree.
the constellations are planar.
We will show in Section 5 that these two conditions translate in fact the planarity of a certain nodal surface.
-Proof of (4.6).We reorganize the terms in (4.1) by the number of excess edges of the abstract graph G " Πpσ, τ q, tπ c u c ; Πpν c q ‰ (i.e. the number of independent cycles, or loop edges in the physics literature) and by the genera of the constellations to get (4.6).
Derivation of (4.1).Our starting point is Theorem 3.3, which states that: where, denoting ν c " σ c τ ´1 c , we have: and ρ denotes the D-uple of constellations pρ 1 , . . ., ρD q where, for c P t1, . . ., Du, the constellation ρc is ρc " pρ c 1 , . . ., ρ c kc q.We aim to derive the asymptotic behaviour of W pN q C rσ, τ s using the results in Sec.2.5.Let us classify the terms in the above formula by the values l c " ř kc ic"1 ρ c ic and by the partitions π c " Πpρ c q ě Πpν c q: We wish to compute The sum ř kě0 p´1q k M pπ, ν; l, kq.Let us focus on the rightmost sum of (4.9).We define: where we emphasize that, contrary to M , the sum defining M includes the case when some of the permutations ρ i are the identity.M and M are related by: M pπ, ν; l, qq " q ÿ k"0 ˆq k ˙M pπ, ν; l, kq , M pπ, ν; l, kq " k ÿ q"0 ˆk q ˙p´1q k´q M pπ, ν; l, qq .
M pπ, ν; l, qq is a sum over q permutations ρ i .The first equation follows by noting that if exactly q ´k out of these permutations are the identity then M pπ, ν; l, qq reduces to M pπ, ν; l, kq; the second equation is obtained by inverting the first one.The point is that M factors over the blocks of π.As ν (respectively ρ i ) stabilizes any block B P π, it can be decomposed as the product of |π| permutations ν |B (respectively , where we lift trivially ν |B (respectively ρ i|B ) to the whole set t1, . . ., nu.The number of transpositions of ρ i is distributed among the blocks of π, ρ i " ř B ρ i|B and we get: where we recognized the number of connected q-constellations with fixed l B , γl B pν |B ; qq of Sec.2.5.
We emphasize that constellations are not necessarily proper (i.e. the sums run over S |B| not S |B| ).The number of arbitrary (i.e.not necessarily proper) constellations is written in terms of the number of proper ones as: and substituting, we find: with the convention that ill-defined binomial coefficients (e.g.k B ą q) are zero.At fixed k B , the sum over q and k can be computed as it is the coefficient of the monomial ś B x k B B in the generating function: which ultimately leads to: Inserting this expression in (4.9) achieves the proof of (4.1): .
This formula can be analyzed further.
The relation between these partitions can be encoded in a convenient graphical representation.Let GrΠ, tπ c u c ; tΠ c u c s be the abstract bipartite graph consisting in: • white vertices associated to the blocks B of the partition Π, • c-colored vertices associated to the blocks B c of the partitions π c • c-colored edges associated to the blocks b c of Π c linking a white and a c-colored vertex.
The block b c is at the same time: contained in a block of Π, which we denote by Bpb c q, as Π ě Π c , -contained in a block of π c , which we denote by B c pb c q, as π c ě Π c .
The edge corresponding to b c links Bpb c q to B c pb c q. Lower bound on l.In order to find a lower bound on l, we first rewrite ř c |π c |. Observe that in (4.1) we sum over partitions tπ c u c such that: @c π c ě Πpν c q , and Πpσ, τ q _ π 1 ¨¨¨_ π D " 1 n .
Since Πpσ, τ q ě Πpν c q, from Lemma 4.2 we conclude that the sum runs over partitions tπ c u c such that the graph G " Πpσ, τ q, tπ c u c ; Πpν c q ‰ is connected.
The graph G " Πpσ, τ q, tπ c u c ; Πpν c q ‰ has ř c |Πpν c q| edges and |Πpσ, τ q| `řc |π c | vertices.If it is connected then any tree spanning this graph will have exactly |Πpσ, τ q| `řc |π c | ´1 edges.We denote the number of excess edges of G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ , that is the number of edges in the complement of a spanning tree in the graph, by: Let us consider a term in (4.1).At fixed l Bc , each block B c of π c contains a sum over constellations with fixed genus as, from (2.11): Summing over c and using n " ν c `|Πpν c q| we get: Thus l ě pσ, τ q leading to Eq. (4.2), which proves (4.3).We see that the bound l " pσ, τ q is attained if and only if g Bc " 0 for all B c , and: which proves (4.4).
Remark 4.3.The condition g Bc " 0 for all B c already suggests that the large-N limit corresponds to some type of planarity.It turns out that the additional condition L " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ " 0 is also a "minimal genus" condition.In Section 5 we will show that the sum in (4.1) can be reinterpreted as a sum over a class of nodal surfaces and at leading order at large-N only nodal surfaces of minimal arithmetic genus contribute.
Formula for l ą " pσ, τ q.More generally for non-minimal l, we can organize the sum in (4.1) by the number of excess edges of the graph G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ .Writing (4.12) as: we reorganize (4.1) as: , by replacing the sum over the partitions tπ c u c with a sum over tπ c u c such that L " ř c `|Πpν c q| |π c | ˘´Πpσ, τ q `1 is fixed and a sum over L from its minimal allowed value 0 to the maximal allowed value pl ´ q{2 fixed by (4.13).Then at fixed L, we use (4.11) and trade the sums over tl c u c , ř c l c " l with sums over the genera g Bc ě 0 constrained by (4.11) to obey: At the end of the day, we get: , where lpg Bc q " |B c | `Πpν c|Bc q `2g Bc ´2, that is (4.6).
This completes the proof of Thm.4.1.

Monotone Hurwitz numbers and their generalization
In D " 1, monotone double Hurwitz numbers are obtained by suming the coefficients p C rσ, τ ; ls for σ, τ of fixed cycle types.These numbers have an interpretation in enumerative geometry.We detail some known facts about monotone Hurwitz numbers, and review the results of our paper in this context.A generalization of monotone double Hurwitz numbers is then introduced, using the coefficients p C for D ą 1.
Monotone double Hurwitz numbers.The definition of the number p C rσ, τ ; ls in Prop.3.6 recalls for D " 1 the combinatorial definition of monotone Hurwitz numbers.It is known that for D " 1 and A and B having asymptotic traces of order N , the HCIZ integral has the expansion [24,25]: Tr β pBq N #pβq `Opt N `1q , (4.15) where α, β are partitions of the integer n, n " ř p p ¨dp pαq, where d p pαq is the number of parts of α of size p, and #pαq " ř p d p pαq denotes the total number of parts of α.Denoting C α the set of permutations having fixed cycle type α, the coefficients: are the genus-h monotone double Hurwitz numbers.In detail these numbers count the number l`2-uplets of permutations σ, τ, µ 1 , . . ., µ l with σ P C α , τ P C β , and µ i transpositions with weakly monotone maxima (Def.2.2) such that σ " µ 1 ¨¨¨µ l τ and furthermore the group generated by σ and all the transpositions acts transitively on t1, . . ., nu.Divided by n!, these numbers also count weighted branched coverings of the Riemann sphere by a surface of genus h with l `2 branch points, l of which have simple ramifications (that is they have n ´1 preimages), and the ramifications profiles at 0 and infinity are given respectively by the partitions α and β (for more details, see Sec. 5.2).The condition that the transpositions have weakly monotone maxima restricts the admissible coverings.The formula relating l (the number of simple branch points) and h is the well known Riemann-Hurwitz formula.Note that if σ " µ 1 ¨¨¨µ l τ and the group generated by σ and all the transpositions acts transitively on t1, . . ., nu, then applying the Euler characteristics formula (2.10) to the constellations pσ, τ ´1q, pµ 1 , . . ., µ l q, and pσ, µ 1 , . . ., µ l q (the last one is connected): h " gpσ, µ 1 , . . ., µ l q " gpσ, τ ´1q `gpµ 1 , . . ., µ l q `L" Πpσ, τ q, π; where π " |Πpµ 1 , . . ., µ l q|, and we recall that L is given by Eq. (4.10).The value of l for h " 0 is fixed by the Riemann-Hurwitz formula as l " #pαq `#pβq ´2 " #pσq `#pτ q ´2 for any pσ, τ q P C α ˆCβ .From Corollary 4.4, this restricts the sum in Eq. (4.16) to σ, τ satisfying gpσ, τ ´1q " 0 and l " pσ, τ q so that: and we can use Thm.4.1 in D " 1 (see also [13]) to express H 0 pα, βq as: Unlike for single Hurwitz numbers (see (4.20) below), one cannot eliminate the sum over permutations, since both the number of connected components and the genus of pσ, τ ´1q depend on the specific representatives pσ, τ q P C α ˆCβ and not only on their conjugacy classes α and β. 6Monotone single Hurwitz numbers.Single Hurwitz numbers are obtained when taking τ " id n (or similarly for σ) above: where we have used the fact that p C rν, id; ls depends only on the partition α induced by the cycles of ν and not on the specific representative ν P C α , since m C pν, id, l, kq is invariant under conjugation (note that we have introduced the somewhat abusive notation p C rα, id n ; ls).The cardinal of C α is |C α | " n!{p ś pě1 p dppαq d p pαq!q.Similarly to (4.15), these numbers are obtained from the HCIZ integral in D " 1, but in the case when the asymptotic moments of the matrix B are degenerate, lim N Ñ8 1 N TrpB k q " δ 1,k [25].Combining Eq. (3.7) and Eq.(4.20) we get that for any h ě 0 and α $ n, the genus-h monotone single Hurwitz numbers are, up to signs, the numbers of connected proper constellations with faces ρ 1 ¨¨¨ρ k " ν P C α , and ř k i"1 ρ i " l for some ν P C α : and this is independent of the representative ν P C α chosen.Like the double numbere, they also count branched coverings of the sphere and requiring the covering to be of genus 0 comes to requiring that l " #pσq `n ´2, that is its minimal possible value pσ, idq.Fixing τ " id n in (4.19), as Πpνq " Πpσ, idq " Πpσq, the sum restricts to π " 1 n and: reproducing the value found in [25,26].
The expressions for single Hurwitz numbers (4.22) were first obtained in [13] as sums over permutations, that is p C expressed7 as in Thm.3.3 and then evaluated for zero genus using the counting of planar constellations of Bousquet-Melou -Schaeffer [5].Higher genus monotone single Hurwitz numbers count higher genus constellations, but for now there is no closed simple formula for them.Single and double monotone Hurwitz numbers were later studied in [24,25,26,27].To our knowledge, result of [13] where ν " στ ´1, and we denoted by cpν B q the cycle type of σ B τ ´1 B (the associated partition of the integer |B|).More generally, using Eq.(4.6) for D " 1 and Eq.(4.21), as well as the fact that if l is given by the Riemann-Hurwitz formula (4.16), then l´ pσ,τ q 2 " h ´gpσ, τ ´1q we get: Theorem 4.5.For any h ě 0 and α, β $ n, the genus-h monotone double Hurwitz numbers are expressed in terms of the single ones as: The sum over partitions can be interpreted as a sum over all ways to add nodes to the 2constellation pσ, τ ´1q with a weight per node given by monotone Hurwitz numbers, as explained in [46] for h " 0 and in Sec. 5 below for h ą 0.
We thus get an expression for monotone double Hurwitz numbers of genus h for partitions of n in terms of the monotone single Hurwitz numbers of genus ď h, for partitions of integers ď n (see e.g.[29,18,1,28]).We do not know if this relation is already known in the literature.
Higher order monotone Hurwitz numbers.The tensor generalization of the HCIZ integral naturally gives rise to the following generalization of monotone Hurwitz numbers: For each color c " 1, . . .D in Prop.3.6, p C rσ, τ ; ls counts factorizations of permutations, that is constellations, and thereby certain branched coverings of genera h c " gpσ c , µ 1 , . . ., µ lc q, with |Πpσ c , µ 1 , . . ., µ lc q| connected components which satisfy the Riemann-Hurwitz formula for nonconnected coverings l c " #pα c q `#pβ c q `2h c ´2|Πpσ c , µ 1 , . . ., µ lc q|.However, the transitivity condition in Prop.3.6, which involves all the permutations, for all c, imposes a global constrained on the coverings.We show in Sec.5.2 that 1 n! H l pα 1 , β 1 , . . ., α D , β D q counts certain weighted connected branched coverings of D distinguishable 2-spheres that "touch" at a single common point.The covering spaces are nodal surfaces, for which a generalization of the genus -the arithmetic genus -is kept fixed.A generalization of the Riemann-Hurwitz formula relates the arithmetic genus to the number of preimages of the branch points.

Interpretation of the combinatorial quantities in terms of nodal surfaces
In this section we give a geometrical picture for the different combinatorial quantities at play: • In Sec.5.2, we interpret transitive factorizations of multiplets of permutations with conditions on the length of the permutations -like m C (Thm.3.3) or p C (Prop.3.6) -as branched coverings of D spheres touching at one common point.
• In Sec.5.3 we develop step-by-step a geometric understanding of m C , p C , and of the combinatorial formula (4.6) that gives an expression for p C in terms of single Hurwitz numbers.
Both geometric descriptions involve nodal surfaces, which are collections of surfaces that "touch" in groups at certain points called nodes.Let us provide more formal definitions.

Nodal surfaces and nodal topological constellations
Nodal surfaces.Given p ě 2 topological spaces X i , 1 ď i ď p, each with a distinguished point x i , the wedge sum of the spaces X i at the points x i is the quotient space of the disjoint union of the spaces X i by the identification @i ă j P t1, . . ., pu, x i " x j .A wedge sum of 2-spheres is often poetically called a bouquet of 2-spheres.
A surface is an orientable manifold of dimension two, together with an orientation.Given a surface X " \ i X i with p connected components X i , as well as r sets of points P j , 1 ď j ď r, such that all the elements in all of the sets P j are distinct points that may belong to any of the connected surfaces X i (a set P j may contain several different points from the same X i ), a nodal surface with r ě 1 nodes is the quotient space of X by the identifications @j P t1, . . ., ru, @x, y P P j , x " y.For each j P t1, . . ., ru, the identification of all the points in P j defines a node or nodal point.Such a nodal surface is then denoted by X ‚ " X{tP j u 1ďjďr , and the X i are said to be its irreducible components.A wedge sum of surfaces is a nodal surface with one nodal point but the converse is not always true: a surfaces may have two or more distinct points in a node.
A nodal surface is said to be connected if for any two points, there exists a path between them, the path being allowed to jump from one surface to another through a nodal point at which they touch.
A map F : X ‚ Ñ Y ‚ between two nodal surfaces is said to be a homeomorphism if: • the restriction of F to each irreducible component of X ‚ is well-defined and is a homeomorphism between surfaces • F preserves the identifications for each node (that is, the points in the irreducible components of the codomain Y ‚ that are identified in a given node are exactly the images of the points that belong to the irreducible components of the domain X ‚ that are identified in a node of X ‚ ).
Arithmetic genus.The arithmetic genus of a connected nodal surface X ‚ " \ p i"1 X i {tP j u 1ďjďr with p irreducible components X i and r nodes is: where gpX i q is the genus of X i and LpX ‚ q is the rank of the first homology group of X ‚ .This is also the number of excess edges of the abstract graph that has a point vertex for each node P j , a square vertex for each X i , and an edge between a point vertex and a square vertex if the corresponding node belongs to the corresponding irreducible component X i , P j X X i ‰ H.The arithmetic genus is the genus obtained by "smoothing" the nodes, whereas ř p i"1 gpX i q is sometimes called the geometric genus of the nodal surface.See the example in Fig. 3, which has geometric genus 2 but arithmetic genus 4 (L " 2).Nodal topological constellations.Consider a D-uplet η " pη 1 , . . ., ηD q of constellations defined on the same set of n elements, where ηc " pη c 1 , . . ., η c kc q is a k c -constellation, k c ě 1.As detailed in Sec.2.5, a graph embedded in a connected surface X is the drawing of a connected graph on X so that the vertices correspond to distinct points on the surface, the images of the edges are paths that may only intersect at the vertices and the complement of the graph in X is homeomorphic to a disjoint union of discs.To simplify the discussion below, we say that a non-connected graph with p components is embedded in a surface with p connected components if the connected components of the graph are embedded in the connected components of the surface.
For X 1 , X 2 two connected surfaces, two embedded graphs Γ 1 Ă X 1 and Γ 2 Ă X 2 are said to be isomorphic if there exists an homeomorphism of surfaces φ : X 1 Ñ X 2 whose restriction to Γ 1 is a graph isomorphism between Γ 1 and Γ 2 .
We consider each constellation ηc as an isomorphism class of (non-necessarily connected) embedded graphs (for more details, see [34]).For each 1 ď c ď D, and for every choice of graph embeddings Γ c Ă X c in the isomorphism class, the white vertices are points on the (non-necessarily connected) surface X c , which we denote by v c i , 1 ď i ď n, and denoting by P j " tv 1 j , . . ., v D j u, we consider the nodal surface \ D c"1 X c {tP j u 1ďjďn , together with the graph Γ c embedded in each surface X c .Two such objects, called here nodal embedded graphs, are said to be isomorphic if there exists an homeomorphism between the nodal surfaces as defined above, such that the restriction to each domain irreducible component is an isomorphism between embedded graphs.
We call nodal topological constellation the resulting isomorphism classes of nodal embedded graphs.It is uniquely encoded by an ordered multiplet of constellations on the same n elements.
Example: the nodal topological constellation Spσ, τ q.We may for instance view pσ, τ ´1q as a nodal topological constellation, which we denote by Spσ, τ q, where the role of ηc is played by the 2-constellation pσ c , τ ´1 c q.We represent for each i P t1, . . ., nu the identification of the white vertices labeled i by introducing a new triangular vertex, linked by dotted edges to the white vertices labeled i in every one of the D bipartite maps pσ c , τ ´1 c q.This is illustrated in Fig. 4.
Isomorphisms and relabeling.Two (topological) k-constellations pη 1 , . . ., η k q and pρ 1 , . . ., ρ k q are said to be isomorphic if thy differ by a relabeling of 1, . . ., n, that is if there exists ν P S n such Figure 4: Graphical representation of Spσ, τ q for an example in D " 2, n " 5, where 2 " p123qp45q.The blue vertices (flavor 1) represent the σs and the red vertices (flavor 2) represent the τ s.Here we have represented the bipartite maps as 2-constellations and added a D-valent triangular node for every i P t1, . . ., nu, between the corresponding white vertices (edges of the bipartite map).In this example, |Πpσ, τ q| " 2, and the arithmetic genus is 2. that η i " νρ i ν ´1 for all i.Two nodal topological constellations encoded respectively by η and ρ where ηc " pη c 1 , . . ., η c kc q and ρc " pρ c 1 , . . ., ρ c kc q are k c -constellations are said to be isomorphic if there exists ν P S n such that for all 1 ď c ď D and all 1 ď i ď k c , η c i " νρ c i ν ´1.Note that ν must be the same for all colors: an isomorphism between nodal constellations is a simultaneous relabeling of 1, . . ., n for all colors.
Transitivity and connectivity.The introduction of nodal surfaces and nodal topological constellations is motivated by the following lemma: Lemma 5.1.For c P t1, . . ., Du, let ηc " pη c 1 , . . ., η c kc q be a k c -constellation, k c ě 1.The number of transitivity classes |Πpηq| of the group generated by all ηc i on t1, . . ., nu is the number of connected components of the corresponding nodal topological constellation.
For instance, Spσ, τ q is connected if and only if the group generated by all σ c , τ c acts transitively on t1, . . ., nu, and more generally, the number of connected components of this nodal topological constellation is |Πpσ, τ q|.
Proof.Consider a representative in the isomorphism class of nodal embedded graphs, that is, a nodal surface X ‚ " \ c X c {tP j u 1ďjďn together with the graph Γ c corresponding to ηc embedded in the surface X c for every c.From the definition of an embedded graph, we know that X c has |Πpη c q| connected components.It is therefore enough to show that two elements a, b P t1, . . ., nu are in the same transitivity class of the group generated by tη c 1 , . . ., η c kc u 1ďcďD if and only if there exists a path between the corresponding nodes on the graph Γ ‚ obtained from Γ 1 , . . ., Γ D by identifying the D white vertices of flavor k for each k P t1, . . ., nu.
Two elements a, b P t1, . . ., nu are in the same transitivity class of the group generated by tη c 1 , . . ., η c kc u 1ďcďD if and only if there exists a word w in these permutations and their inverses so that wpaq " b.Assuming that this is the case, we may build a path between the points corresponding to the two nodes labeled a and b in t1, . . ., nu in Γ ‚ as follows: we read the word w from right to left, when encountering a permutation η c i pdq, 1 ď d ď n the path follows the two edges of flavor i from the node labeled d to the node labeled η c i pdq on the embedded graph Γ c Ă X c , and similarly for η c i pdq ´1.The important point is that there is no problem in successively applying permutations of different colors, since the path may go between any two Γ c Ă X c and Γ c 1 Ă X c 1 at any node.Conversely, a path in Γ ‚ from a node a to a node b is composed of successive steps from a node to a vertex of flavor i via an edge e and on to another node j via an edge e 1 is some Γ c for some c.To each such step we associate the permutation pη c i q pd`1q , d being the number of edges encountered when turning from e to e 1 around the flavored vertex clockwise.A word w such that wpaq " b is then obtained by composing these permutations from right to left.Now, as the number of connected components of the nodal topological constellation is |Πpηq|, the number of its irreducible components is ř c |Πpη c q|, its geometric genus is ř c gpη c q, and the surface has n nodal points each with cardinal D, we obtain the arithmetic genus (5.1) of the nodal topological constellation: For instance, combining this formula for Spσ, τ q, with Eq. (2.11) for the Euler characteristics of pσ c , τ ´1 c q we get: ´#pσ c q ´#pτ c q . ( Supplementing this by the definition (4.2) pσ, τ q " ř D c"1 σ c τ ´1 c `2`| Πpσ, τ q| ´1˘l eads to the following lemma.

Transitive factorizations of multiplets of permutations and branched coverings of a bouquet of 2-spheres
Given two topological spaces X and Y , and a subset L of Y , a map f : X Ñ Y is said to be an n-sheeted branched covering of Y branched over L, if f restricted to the complement of the preimage of L in X is continuous, and such that for every y P Y zL, there exists an open neighborhood U such that f ´1pU q is homeomorphic to U ˆt1, . . ., nu.Two branched coverings f 1 : X 1 Ñ Y and f 2 : X 2 Ñ Y are said to be isomorphic if there exists an orientation preserving homeomorphism u : X 1 Ñ X 2 such that f 1 " f 2 ˝u.The set L is called the branch locus, Y the target space, and X the covering space.The number of connected components of a covering is that of the covering space.For X, Y two nodal surfaces, L consists of points called branch points, and their preimages are called singular points.
It is well known (see [34]) that n-sheeted coverings of the oriented 2-sphere branched over k ordered points up to isomorphisms are in one-to-one correspondence with k-constellations η " pη 1 , . . ., η k q, that is k-uplets of permutations of n elements, such that η 1 ¨¨¨η k " id, up to isomorphisms.Given such an unlabelled constellation, an isomorphism class of branched coverings is obtained by sending each face of the corresponding topological constellation (for every surface in the isomorphism class) to the face of the unique constellation with one white vertex.Each "star" in the constellation formed by a white vertex and its incident edges thus corresponds to the preimage of the only "star" in the target space.The vertices with flavors of the constellation correspond to the singular points, and the partitions of n that label the conjugacy classes of the permutations η 1 , . . ., η k , called ramification profiles, describe the way in which the n sheets meet in groups at the singular points.The covering space is a collection of K " |Πpηq| connected surfaces seen up to isomorphisms, whose genera sum up to h " gpηq.The Riemann-Hurwitz formula relates these two numbers: where for the branch point labeled i, η i " n ´#pη i q is the difference between the number n of preimages that the point would have if it was not in the branch locus, and the number of preimages it actually has.A D-uplet of constellations η " pη 1 , . . ., ηD q, where ηc " pη c 1 , . . ., η c kc q is a k c -constellation on n elements, k c ě 1, up to isomorphisms is therefore in bijection with D branched coverings f 1 , . . .f D of the 2-sphere S, up to isomorphisms, f c : X c Ñ S being branched over k c points.Unlike for nodal constellations, here the isomorphisms are for each color independently, that is, independent relabelings of 1, . . ., n for different colors are allowed.There is no direct interpretation in this context for the quantity |Πpηq|, which moreover is not invariant under relabelings of t1, . . ., nu for each color independently: it is only invariant under simultaneous relabelings for all colors.On the other hand, |Πpηq| has a natural interpretation in the context of nodal constellations, as stated in the following theorem.• For c P t1, . . ., Du, η c 0 , . . ., η c kc`1 P S n such that we have id " η c 0 ¨¨¨η c kc`1 , • |Πpηq| " 1, that is, the group generated by all the permutations is transitive on {1,. . ., n}, up to isomorphisms of nodal constellations (up to simultaneous relabeling of 1, . . ., n for all 1 ď c ď D).
Proof.We prove the correspondence between topological objects, knowing the correspondence between nodal topological constellations and systems of permutations.Consider a branched covering f : X ‚ Ñ Y ‚ where Y ‚ is a bouquet of D distinguishable 2-spheres S c , c P t1, . . ., Du.
On each 2-sphere S c of the target space Y ‚ , one can draw a star-graph γ c by adding non-crossing arcs between the k c branch points and the nodal point so that the order of the arcs around the nodal point grows from 1 to k c clockwise (see the right of Fig. 5).Doing this for all c, we get a nodal embedded graph γ ‚ Ă Y ‚ , whose preimage Γ ‚ Ă X ‚ is a representative of a nodal topological constellation in the sense that it is a representative in the corresponding isomorphism class of nodal embedded graphs.There is no labeling of the n preimages of the nodal point, so that the nodal constellation can be seen up to isomorphisms (up to simultaneous relabelings of 1, . . ., n for all 1 ď c ď D).
For two isomorphic branched coverings f 1 : X ‚ 1 Ñ Y ‚ and f 2 : X ‚ 2 Ñ Y ‚ , there exists by definition an orientation preserving homeomorphism of nodal surfaces u : 1 is an isomorphism between embedded graphs: it is an homeomorphism of surfaces by definition, and it is clear that the restriction of u to Γ ‚ 1 on each irreducible component is a graph isomorphism.Therefore, the nodal embedded graphs Γ ‚ 1 Ă X ‚ 1 and Γ ‚ 2 Ă X ‚ 2 are isomorphic, and are two representatives of the unlabelled nodal topological constellation.
This defines a map from isomorphisms classes of branched coverings of Y ‚ to isomorphism classes of nodal topological constellations, and we now verify that this map is invertible.Indeed, consider a representative Γ ‚ Ă X ‚ of a nodal topological constellation η " pη 1 , . . ., ηD q, where ηc " pη c 1 , . . ., η c kc q is a k c -constellation.Γ ‚ Ă X ‚ is a nodal embedded graph, and for every c, we denote by X c the disjoint union of the irreducible components of X ‚ that contain vertices associated with η c 0 .A branched covering f : X ‚ Ñ Y ‚ is then obtained by choosing homeomorphisms sending each connected component of the complement of the graph Γ ‚ in X c to the complement of the star-graph γ c in the irreducible component S c of Y ‚ .
Given two representatives Γ ‚ 1 Ă X ‚ 1 and Γ ‚ 2 Ă X ‚ 2 of a nodal topological constellation, there exists an homeomorphism of nodal surfaces u : X ‚ 1 Ñ X ‚ 2 that induces an isomorphism of embedded graphs on every irreducible component of X ‚ 1 .Considering the branched coverings as in the previous paragraph, we see that f 1 " f 2 ˝u so that f 1 and f 2 are isomorphic.
The construction described above that associates a covering f to a representative Γ ‚ Ă X ‚ is independent of the labeling of the nodal points of Γ ‚ , so that we have defined the converse map from isomorphism classes of nodal topological constellations to isomorphisms classes of branched coverings of Y ‚ .
The statement regarding the number of connected components is a direct consequence of Lemma 5.1.
We illustrate this for the following example with n " 5, D " 2, k 1 " 2, k 2 " 3, for which the nodal topological constellation is represented graphically on the left of Fig. 5: For the permutations of η1 : η 1 0 " p12qp3qp4qp5q, η 1 1 " p12qp34qp5q, η 1 2 " p12qp345q, η 3 1 " p12qp3qp45q; For the permutations of η2 : η 2 0 " p132qp45q, η 2 1 " p15qp24qp3q, η 2 2 " p1qp23qp4qp5q, η 2 3 " p14qp2qp35q, η 4  2 " id 5 .In the figure, the color representing the flavors 0,1,2,3,4 are in that order pink, blue, red, orange, green.The nodal topological constellation is connected and has arithmetic genus 4. The fact that η 4  2 " id 5 (all green vertices are leaves in the nodal constellation) means that when interpreted as a branched covering of a bouquet of two 2-spheres, the green vertex in the target space is actually not a branch point since it has 5 preimages.Fixing D, k ě 1, H ě 0 and for all c P t1, . . ., Du, α c , β c $ n non-trivial, we let C H rtα c , β c u c , ks be the set of isomorphism classes of connected n-sheeted branched coverings of a bouquet of D distinguishable 2-spheres S c , c P t1, . . ., Du branched over a set of precisely8 k `2D ordered points that do not belong to the nodal point, at least two of which belong to S c for each c, so that the first and last points for each c respectively have ramification profiles α c and β c9 , and so that the arithmetic genus of the covering space as defined in (5.1) is H.
Note that for an element X of C H rtα c , β c u c , ks, the nodal surfaces in the isomorphism class have n nodal points (the node of the bouquet of spheres does not belong to the branch locus and its n preimages are the only nodes of X), so that LpXq " npD ´1q ´p `1 in (5.1).
We let B H rtα c , β c u c , ks be the subset of C H rtα c , β c u c , ks of the elements X for which the branch points whose ramification profiles are not fixed to one of the α c or β c have simple ramification (they have n ´1 preimages) and satisfy an additional monotonicity condition: Consider any set of permutations encoding X (a choice of labeling of 1, ..., n in Th. 5.3).For c P t1, . . ., Du, the transpositions encoding the ramification profiles of the branch points in S c whose ramification profiles are not fixed to α c or β c inherit an ordering from the global ordering of the branch points.With this ordering, these transpositions must have weakly monotone maxima.
We recall that for α $ n, the conjugacy class C α gathers the permutations in S n whose cycle-type is α, that m C and p C were respectively defined in Thm.3.3 and Prop.3.6, as well as the definition of higher order monotone double Hurwitz numbers (4.24): We also define the following generalization of the Bousquet-Melou-Schaeffer numbers [5]: 2. The cardinal of B H rtα c , β c u c , ks is 1 n! H k ptα c , β c u c q and k " l as defined in (5.8).
In both cases, the total number of singular points is given by: nk ´l `ÿ c #pα c q `#pβ c q " 2 ´2H `npk `D ´1q. (5.9) The relation (5.8) should be compared to the Riemann-Hurwitz formula (4.16).
Proof.Let f : X ‚ Ñ Y ‚ be an element of C H rtα c , β c u c , ks, where Y ‚ is a bouquet of D distinguishable 2-spheres S c , c P t1, . . ., Du.Then there exists k 1 , . . ., k D ě 0 such that ř c k c " k and for each c, k c `2 of the (ordered) branch points belong to S c , and the first and last respectively have ramification profiles α c and β c .From Thm. 5.3 and its proof, f is bijectively mapped to D ordered sequences of permutations: ηc " pσ ´1 c , η c 1 , . . ., η c kc , τ c q P S kc`2 n , s.t.id " σ ´1 c η c 1 ¨¨¨η c kc τ c , c P t1, . . ., Du, such that σ c P C αc and τ c P C βc , and the group generated by all the permutations is transitive on t1, . . ., nu, up to simultaneous relabelings of 1, . . ., n for all c, and so that the arithmetic genus of the nodal topological constellation encoded by this system of permutations is is given by the right hand side of (5.8).From (5.2), H " ř D c"1 pgpη c q ´|Πpη c q|q `npD ´1q `1, and from the Euler characteristics (2.11) of ηc , 2 pgpη c q ´|Πpη c q|q " ř kc i"1 η c i ´#pσ c q ´#pτ c q, so that: r#pσ c q `#pτ c qs `2pH ´npD ´1q ´1q, which proves the first point of the corollary.For the elements of B H rtα c , β c u c , ks, η c 1 , . . ., η c kc are transpositions with weakly monotone maxima, and the total number k of these transposition is also ř D c"1 ř kc i"1 η c i .This concludes the proof.
Remark 5.5.For the case D " 2, the enumeration of isomorphism classes of branched covers of a bouquet of two 2-spheres should be relevant in the context of compactifications of the moduli spaces of curves such as the Deligne-Mumford compactification, where the necessity to include degenerated cycles implies considering nodal surfaces where at each node only two surfaces meet [21,11,47,35].
Remark 5.6.We have presented a geometrical interpretation based on nodal surfaces.From the colored structure, the reader familiar with the literature on colored triangulations and random tensor models will recognize a combinatorial encoding that recalls that of colored triangulations in dimension two and higher.This begs for an interpretation in terms of higher dimensional objects, instead of nodal surfaces, but we leave this for future work.

The 1{N expansions as topological expansions
The aim of this subsection is to provide a combinatorial and geometric interpretation to the formulas of Theorem 4.1.The transpositions µ c i in the combinatorial definition of p C (Prop.3.6) do not appear for instance in (4.6): the intuition is that we should try to keep all the σ c , τ c fixed on one hand, and "resum" the contributions of all the µ c i on the other hand, in some way.To this aim, given D sequences of permutations σ c , τ c , η c 1 , . . ., η c kc for c P t1, . . ., Du such that id " σ ´1 c η c 1 ¨¨¨η c kc τ c , instead of considering the nodal topological constellation encoded by the pσ ´1 c , η c 1 , . . ., η c kc , τ c q for all c as in Corollary 5.4, we will rather consider a new kind of isomorphism class of nodal surfaces from the nodal topological constellation Spσ, τ q on one hand, and the D topological constellations ηc " pη c 1 , . . ., η c kc q on the other hand.
(5.10) subject to the conditions: (C1) the collection of all tη c , τ c u c acts transitively on t1, . . ., nu, This data defines: • a (non-necessarily connected) nodal topological constellation Spσ, τ q as defined in Sec.5.1 (see Fig. 4), • a (non-necessarily connected) topological k c -constellation ηc for each c P t1, . . ., Du. Since and the disjoint cycles of η c 1 ¨¨¨η c kc match, so that for every nodal embedded graph Γ ‚ Ă X ‚ in the isomorphism class Spσ, τ q and every embedded graphs Γ 1 Ă Y 1 , . . ., Γ D Ă Y D in the isomorphisms classes η1 , . . ., ηD , there is a one-to-one correspondence Ψ between the faces F 1 , . . ., F nk´l of Γ ‚ Ă X ‚ (the connected components of the complement of the graph Γ ‚ in the nodal surface X ‚ ), and the faces F 1 1 , . . ., F 1 nk´l of the Γ c Ă Y c for c P t1, . . ., Du, where the labelings are chosen so that ΨpF j q " F 1 j .To render this pairwise identification obvious, we choose for each j P t1, . . ., nk ´lu two points v j and v 1 j respectively in the interiors of F j and F 1 j , and we consider the nodal surface Z ‚ " pX ‚ \ D c"1 Y c q{tP j u j (together with the graphs Γ ‚ and Γ c drawn on Z ‚ ).
We then call Spσ, τ , ηq the isomorphism class of such objects, where by isomorphisms we mean the homeomorphisms of nodal surfaces that induce an isomorphism of embedded graph on each irreducible component, and preserve the incidence between the nodes P j and the faces F j and F 1 j , in the sense that if a node P j belongs to the interior of the faces F j and F 1 j , then the image of the node also belongs to the interior of the images of the faces.
An example is shown in Fig. 6, where the nodal points in the interior of the faces are represented by dotted edges (whereas we recall that the nodes of Spσ, τ q are represented by dashed edges linking triangular vertices).
In this context, the graph G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ for π c " Πpη c q and ν c " σ c τ ´1 c introduced in Sec.4.2 is simply obtained by contracting the connected components of the nodal surface Spσ, τ q (not its irreducible components!) and those of each constellation ηc to points.This retains the information on which faces F j and F 1 j identified by Ψ are in the same connected component of Spσ, τ q on one hand, and which ones are in the same connected component of ηc on the other (this is why it only depends on the associated partitions).The graph G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ for the example in Fig. 6 is the one in Fig. 2.

2.
The nodal topological constellations encoded by η1 " p η1 1 , . . ., η1 D q where η1 c " pσ ´1 c , ηc , τ c q, and in both cases, the spaces are connected and of fixed arithmetic genus G ě Gpσ, τ q, and so that the vertices of flavor i of the ηc are not all of valency one.p C rσ, τ ; ls counts the subset of the spaces listed above for which the ηc consist of sequences of transpositions with weakly monotone maxima (this also translates on a condition on the flavored vertices).
The arithmetic genus Gpσ, τ , ηq, or equivalently the exponent l " ř D c"1 ř kc ic"1 η c ic , can then be expressed as (Sec.4.2): Gpσ, τ , ηq ´Gpσ, τ q " l ´ pσ, τ q 2 " D ÿ c"1 gpη c q `L" Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ , ( where gpη c q is the genus of the k c -constellation ηc (the sum of genera of its connected components) and L " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ the number of excess edges (4.10) of G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ : L " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ " ÿ c p|Πpν c q| ´|π c |q ´|Πpσ, τ q| `1. ( . This provides a better understanding on how to characterize and count the contributions to m C and p C in Prop.5.9: counting connected Spσ, τ , ηq of fixed arithmetic genus G amounts to counting those Spσ, τ , ηq for which the graph G " . . .‰ is connected and has excess L between 0 and G ´Gpσ, τ q while the genera gpη c q sum up to G ´Gpσ, τ q ´L (the other conditions in Prop.5.9 must also be satisfied).In the example of Fig. 6, the constellations are planar and pl ´ q{2 " 2 " L. For instance, the following gives a prescription for generating all the spaces that contribute at leading order: Proposition 5.10.The spaces Spσ, τ , ηq that correspond to the leading term in N (4.3) for the expansion of the cumulant Weingarten functions W pN q C rσ, τ s are those of minimal arithmetic genus Gpσ, τ , ηq " Gpσ, τ q, that is, so that the ηc are all planar and so that G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ is a tree.
From this picture the aim is to keep Spσ, τ q fixed and group the contributions of the different constellations η that lead to the same values of L and the same genera for the connected components of the ηc .This is achieved in the last subsection.

A simpler kind of nodal surfaces
Both for m C and p C , one fixes σ, τ but sums over the proper η (denoted by ρ or μ) satisfying a number of assumptions.In order to understand this geometrically, one may therefore, from Spσ, τ ; ρq introduced in the previous subsection, contract to points the connected components of the constellations ηc but keep the nodal topological constellation Spσ, τ q (that is go only half-way to build the graph G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ ).The result is a new kind of object Spσ, τ ; tπ c u c q (since only the information on the connected components of the constellations is retained, they have been replaced in the argument by the corresponding partitions on t1, . . ., nu).It can be obtained directly from Spσ, τ q by adding nodal points between the faces of Spσ, τ q corresponding to the blocs of tπ c u c .Let us introduce this object more formally.We fix σ, τ P S n as well as G ě 0, and for c P t1, . . ., Du, we let π c ě Πpν c q be a partition of the disjoint cycles of ν c " σ c τ ´1 c subject to the conditions: (C'1) Πpσ, τ q _ π 1 _ . . ._ π D " 1 n , (C'2) L " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ " G ´Gpσ, τ q.
For every nodal embedded graph Γ ‚ Ă X ‚ in the isomorphism class Spσ, τ q, tπ c u c provides a partition of the faces F 1 , . . ., F nk´l of Γ ‚ Ă X ‚ (the connected components of the complement of the graph Γ ‚ in the nodal surface X ‚ ).We may see the blocks in this partition as a new kind of node: we choose for each j P t1, . . ., nk ´lu a point v j in the interior of F j , and we see each block B c of π c for each c as a node P pB c q " tv j | F j P B c u.We then consider the nodal surface X ‚ { \ c tP pB c qu BcPπc (together with the nodal graph Γ ‚ embedded on X ‚ ).
We then denote by Spσ, τ ; tπ c u c q the isomorphism class of such objects, where by isomorphisms we mean the homeomorphisms of nodal surfaces that induce an isomorphism of nodal embedded graph on Γ ‚ Ă X ‚ and that preserve the incidence between the nodes P pB c q and the faces F j , in the sense that if a node P pB c q belongs to the interiors of the faces F j P B c , then the image of the node also belongs to the interior of the images of the faces.We let: S G pσ, τ q " !Spσ, τ ; tπ c u c q such that tπ c u c satisfy (C'1) and (C'2) and more generally, we call the Spσ, τ ; tπ c u c q the foldings of Spσ, τ q.
An example is shown in Fig. 7, where now the nodal points in the interiors of the faces are represented by dotted edges that meet at star-vertices labelled by the blocks of the partitions π c .
As for Lemma 5.7, the following follows directly from the fact that G " Πpσ, τ q, tπ c u c ; tΠpν c qu c ‰ is obtained by contracting the connected components of Spσ, τ q to points: The topological expansion of the cumulant Weingarten functions in Prop.5.9 can therefore also be re-expressed as a topological expansion over connected foldings of Spσ, τ q of fixed arithmetic genus.
With this interpretation, generating all the contributions to p C rσ, τ ; ls is quite simple: fixing σ, τ and l, one sums over the excess L between 0 and pl ´ q{2, and over all possible ways to add nodes of color c (represented by star-vertices of color c) for every color c P t1, . . ., Du between all the faces of Spσ, τ q, so that the resulting (class of) nodal surface is connected and the graph obtained when contracting the connected components of Spσ, τ q to points has L excess-edges.This generates all the foldings of Spσ, τ q of arithmetic genus Gpσ, τ q `L.For each such folding S " Spσ, τ ; tπ c u c q, each node of color c corresponds to a block B c of π c .One then distributes the total genus l´ 2 ´L (see (5.13)) among all the nodes of color c, and each such node is endowed with a weight γ lpg Bc q pν c|Bc q, that precisely takes into account the contributions of all the connected constellations of genus g Bc corresponding to the connected component of ηc for every Spσ, τ , ηq that contracts to S. From (4.21) this factor γ lpg Bc q pν c|Bc q is proportional to a genus-g Bc single monotone Hurwitz number (the signs combine in the overall p´1q l factor in (4.6)).
The simplified version of this geometrical picture corresponding to D " 1 and minimal l " has been introduced in [46].

Figure 3 :
Figure 3: A nodal surface of arithmetic genus 4 (left), and a surface of genus 4 obtained by "smoothing" the nodes (right).

Theorem 5 . 3 .
Isomorphism classes of connected branched coverings of a bouquet of D distinguishable 2-spheres S c , c P t1, . . ., Du branched over a set of k `2D ordered points that do not belong to the nodal point, k c `2 of which belong to S c for each c ( ř D c"1 k c " k) are in one-to-one correspondence with systems of permutations of the type:

1 S 2 Figure 5 :
Figure5: A nodal topological constellation (left) can be interpreted, up to relabeling of the nodes, as as an isomorphism class of branched coverings f of a bouquet of D distinguishable 2-spheres (right).In this example, D " 2, n " 5, k 1 " 2, and k 2 " 3.
leading to Eq. (4.19) is the only explicit expression of monotone double Hurwitz numbers of genus 0. Double numbers in terms of the single ones.Note that equation Eq. (4.19) expresses monotone double Hurwitz numbers of genus zero in terms of monotone single Hurwitz numbers::