Pruned double Hurwitz numbers

Hurwitz numbers count ramified genus $g$, degree $d$ coverings of the projective line with with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over $0$ and $\infty$ and only simple ramification else. These objects feature insteresting structural behaviour and connections to geometry. In this paper, we introduce the notion of pruned double Hurwitz numbers, generalizing the notion of pruned simple Hurwitz numbers in \cite{DN13}. We show that pruned double Hurwitz numbers, similar to usual double Hurwitz numbers, satisfy a cut-and-join recursion and are piecewise polynomial with respect to the entries of the two special ramification profiles. Furthermore double Hurwitz numbers can be computed from pruned double Hurwitz numbers. To sum up, it can be said that pruned double Hurwitz numbers count a relevant subset of covers, leading to considerably smaller numbers and computations, but still featuring the important properties we can observe for double Hurwitz numbers.


Introduction
Hurwitz numbers are important enumerative objects connecting numerous areas of mathematics, such as algebraic geometry, algebraic topology, operator theory, representation theory of the symmetric group and combinatorics. Historically, these objects were introduced by Adolf Hurwitz in [13] to study the moduli space ℳ of curves of genus . There are various equivalent definitions of Hurwitz numbers and several different settings, among which the most well-studied one is the case of simple Hurwitz numbers, which we denote by ℋ ( ). To be more precise, simple Hurwitz numbers count genus coverings of P 1 (C) with fixed ramification over 0 and simple ramification over further fixed branch points, where the number is given by the Riemann-Hurwitz formula. The theory around these objects is well developed and a lot is known about their structure. Each degree cover : → with branch locus induces a monodromy representation, i.e. a map : 1 ( ∖ ) → S . Starting from these monodromy representations and applying Riemann's existence theorem one can show that there is an equivalent definition in terms of factorizations of permutations (see chapter 7.2 in [5]). Moreover, simple Hurwitz numbers satisfy a cut-and-join recursion which is inherent in the combinatorial structure of these factorizations. Another well-known result is the fact that -up to a combinatorial factor -ℋ ( ) behaves polynomially in the entries of for fixed genus and fixed length of . Recently, there has been an increased interest in Hurwitz theory due to connections to Gromov-Witten theory, remarkably through the celebrated ELSV formula [8] which relates Hurwitz numbers to intersection products in the moduli space of curves. This formula initiated a rich interplay between those areas. The polynomiality result for simple Hurwitz numbers is a consequence of the ELSV formula. Via the ELSV formula a new proof of Witten's conjecture was given in [17] using Hurwitz theory. Moreover, simple Hurwitz numbers satisfy the Chekhov-Eynard-Orantin topological recursion, a theory motivated by mathematical physics with numerous applications in geometry (see e.g. [3], [1], [9], [10]).
A further case which has been of great interest in recent years is the one of double Hurwitz numbers, which we denote by ℋ ( , ). Here we allow two special ramification profiles, that is in addition to allowing arbitrary ramification over 0, we allow arbitrary ramification over ∞. Obviously, for = (1, . . . , 1) this yields the definition for simple Hurwitz numbers given above. While there are still a lot of open questions, much is known about these objects as well and they admit many results, which are similar to those about simple Hurwitz numbers. Among those is a cut-and-join recursion for double Hurwitz numbers and a definition in terms of factorizations in the symmetric group. In [12] it was proved that ℋ ( , ) behaves piecewise polynomially in the entries of and . More than that, wall-crossing formulas in genus 0 were given in [18] and in all genera in [4] and [15]. Among the open problems for double Hurwitz numbers is the question, if there is an ELSV-type formula for them [2]. Some progress has been made in [12], where such a formula is given for genera 0 and 1. Furthermore, it is not known, whether double Hurwitz numbers satisfy an Eynard-Orantin topological recursion.
In [6] the notion of pruned simple Hurwitz numbers was introduced. The main idea behind this notion is, that it is sufficient to consider a non-trivial subset of ramified covers that contribute to the simple Hurwitz number that still carries all the information and that this subset may be described purely combinatorially in terms of certain graphs on surfaces. These graphs were introduced as branching graphs in [17]. There are various names in the literature for these and similar graphs, such as ribbon graphs, dessin d'enfants, Hurwitz galaxies, maps in surfaces, graphs in surfaces. The pruned simple Hurwitz number, which we denote by ℋ ( ) is a count over this subset. It was established in [6], that simple Hurwitz numbers and pruned simple Hurwitz numbers are equivalent in the sense, that simple Hurwitz numbers may be computed as a weighted sum over certain pruned simple Hurwitz numbers of the same genus. Moreover, these new objects still carry a lot of information of the standard case, such as the fact that ℋ ( ) behaves polynomially in the entries of . Pruned simple Hurwitz numbers are defined in terms of graphs on surfaces, however there is a definition in terms of factorizations of permutations, as well. Moreover, they admit a cut-and-join recursion similar to the one for simple Hurwitz numbers. Using these results and the ELSV formula, another proof for Witten's Conjecture was given in [6]. Furthermore, it was proved, that pruned simple Hurwitz numbers admit an Eynard-Orantin topological recursion.
To sum up, it can be said that simple pruned Hurwitz numbers count a relevant subset of covers, leading to considerably smaller numbers and computations, but still featuring the important properties we can observe for simple Hurwitz numbers.
The aim of this paper is to introduce the notion of pruned double Hurwitz numbers, generalizing the definition in [6] and to investigate their structure. Our definition of pruned double Hurwitz numbers, which we denote by ℋ ( , ), is given in terms of branching graphs, as well. We prove three structural results about pruned double Hurwitz numbers: Theorem 1. Double Hurwitz numbers can be expressed in terms of pruned double Hurwitz numbers with smaller input data (i.e. smaller degree and ramification data, but the same genus).
For a precise formulation see Theorem 15. Section 3 is devoted to the proof of this theorem.

Theorem 2. Pruned double Hurwitz numbers satisfy a cut-and-join recursion.
For a precise formulation see Theorem 24, which is proved in Section 4. For a precise formulation see Theorem 31, which is proved in the first half of Section 5. Moreover, we express pruned double Hurwitz numbers in terms of factorizations in the symmetric group. We begin this paper by recalling some basic facts about Hurwitz numbers and re-introducing branching graphs in a way suitable for our purposes in Section 2. In Section 3, we introduce the notion of pruned double Hurwitz numbers and prove Theorem 1. We continue in Section 4 by formulating and proving Theorem 2. In Section 5, we give a proof for Theorem 3. We note, that while our first two results are proven in a similar way as their corresponding results in [6], the method used for the polynomiality result is not feasible for pruned double Hurwitz numbers. In fact, our method is similar to the one used in [12] to prove the piecewise polynomiality for double Hurwitz numbers. We finish this section by connecting the combinatorics of branching graphs to the combinatorics of symmetric groups and express pruned double Hurwitz numbers in the setting of factorizations of permutations. Building on these results, we developed and implemented an algorithm to compute pruned double Hurwitz numbers. An implementation of the algorithm in the computer algebra system [11] may be found in https://sites.google.com/site/marvinanashahn/computer-algebra. Using this tool, we computed several non-trivial examples of Hurwitz numbers and pruned Hurwitz numbers. The computations agree with the predictions made by the formulas of Theorem 15 and Theorem 24.

Preliminaries
In this section, we introduce some basic notions of graph theory and the theory of Hurwitz numbers. Detailed introductions to these topics can be found in [19], [16] p.84-92 and the book [5].

Graphs
We consider graphs with half edges ( , , ′ ). Here is the set of vertices and the multiset ⊂ × is the set of edges. The multiset ′ ⊂ is the set of half-edges. A forest is a graph without cycles and a tree is a connected forest.
We note, that we define the valency val( ) to be the number of full-edges incident to . By convention, we count loops twice. Obviously, we may decompose each graph into its connected components. We call a forest rooted, if each component contains a distinguished vertex, which we call the rootvertex. Note that a rooted forest carries a canonical orientation in the way, that the edges of each connected component point away from the corresponding root-vertex (see e.g. Figure 2).
We call a branch point with ramification profile (2, 1, . . . , 1) a simple branch point and we call a ramification point with ramification index 2 a simple ramification point. An isomorphism between two covers : → P 1 (C), ′ : ′ → P 1 (C) is a homeomorphism : → ′ respecting the labels, such that ′ ∘ = . We denote the automorphism group of a cover by Aut( ). Let H ( , ) be the set of all Hurwitz covers of type ( , , ). Then we define the double Hurwitz number Note that ℋ ( , ) is a topological invariant, that is, it is independent of the locations of the points 1 , 2 , 1 , . . . , and of the complex structure of .
By matching a cover with a monodromy representation, we may count ramified coverings of P 1 (C) in terms of factorizations of permutations. For a permutation , denote by ( ) the corresponding partition given by its decomposition in disjoint cycles.
Proof. For a proof, see for example [16].

Hurwitz galaxies and Branching graphs
In this subsection, we explain a connection between covers contributing to ℋ ( , ) and graphs on surfaces. We will define two notions of graphs on surfaces, that will turn out to be equivalent. We will start by defining branching graphs. We note that we will view full-edges as two half-edges glued together at their respective vertex-free ends.
Definition 7. Let be a positive integer, and be ordered partitions of . We define a branching graph of type ( , , ) to be a graph embedded on an oriented surface of genus g, such that for = 2 − 2 + ℓ( ) + ℓ( ): (i) ∖ is a disjoint union of open disks.
(iv) The ℓ( ) faces are labeled by 1, . . . , ℓ( ) and the face labeled has perimeter ( ) = , by which we mean, that each label occurs times inside the corresponding face, where we count full-edges adjacent to on both sides twice.
Note that we allow loops at the vertices. An isomorphism between two Hurwitz galaxies, is an orientation-preserving homeomorphism of their respective surfaces, which induces an isomorphism of graphs, that preserves vertex-, (half-)edge-and face-labels. Now we will define a second notion of graphs on surfaces, namely Hurwitz galaxies (see e.g. [7] or [14]). (iii) these faces may be coloured black and white, such that ℓ( ) many faces are coloured black and ℓ( ) many faces are coloured white, such that each edge is incident to a white face on one side and to a black face on the other side, (iv) the white (resp. black) faces are labeled by 1, . . . , ℓ( ) (resp. 1, . . . , ℓ( )), such that a face labeled is bounded by · vertices, An isomorphism between two Hurwitz galaxies is an orientation-preserving homeomorphism of their respective surfaces, which induces an isomorphism of graphs, that preserves vertex-and face-labels. Proposition 9. There is a bijection:
Proof. We start with a Hurwitz galaxy of type ( , , ). Draw a vertex in each white face und connect this vertex to the vertices surrounding this face. Now remove the vertices of the old graph . We obtain a branching graph of type ( , , ) by distributing the labels naturally. Obviously, we may reverse this process and thus get the bijection as desired.
Example 10. We illustrate the construction in the proof of Proposition 9 in Figure 3. We start with a Hurwitz galaxy of type (0, (2, 1, 3), (1, 2, 1, 2)) and obtain the corresponding branching graph of type (0, (2, 1, 3), (1, 2, 1, 2)). The green numbers display the labels of the faces of the galaxy and the labels of the faces and vertices of the branching graph.
We will construct Hurwitz covers from branching graphs . In this construction, we will actually use Hurwitz galaxies . Moreover, we want to relate the automorphism groups. To be more preice, we will see that there are natural bijections between the set of Hurwitz covers of type ( , , ), branching graphs of type ( , , ) and Hurwitz galaxies of type ( , , ). Furthermore, we will see that for a Hurwitz cover , the corresponding branching graph and Hurwitz galaxy , there are natural isomorphisms between their the electronic journal of combinatorics 22 (2015), #P00  Figure 4: On the left, graph on the sphere, whose pullback yields a Hurwitz galaxy. On the right, graph on the sphere, whose pullback yields a branching graph.
automorphism groups. We note that only branching graphs of type ( , ( ), ( )) have automorphisms. This may be seen by an easy graph theoretic argument. We will give a proof by connecting the automorphisms of branching graphs to automorphisms of factorizations in the symmetric group in Section 5.
We can compute Hurwitz numbers in terms of isomorphism classes of branching graphs of type ( , , ). We denote the set of all isomorphism classes of branching graphs of type ( , , ) by ℬ ( , ) .
Proposition 11 ( [17], [12], [14]). With notation as above, we have: The idea behind the proof of Proposition 11 is to express Hurwitz galaxies and branching graphs as pullbacks of certain graphs on P 1 (C) in the following sense: Fix some ∈ H ( , ). Draw the graph whose vertices are the = 2 − 2 + ℓ( ) + ℓ( ) roots of unity and whose edges connect them as in the left graph in Figure 4. The pre-image of this graph under is a Hurwitz galaxy of type ( , , ) and each Hurwitz galaxy of type ( , , ) appears that way. Similar for branching graphs, we draw the graph whose vertices are the roots of unity and 0 on P 1 (C) and whose edges connect 0 to each root of unity as in the right graph in Figure 4 and take the pre-image.

Pruned double Hurwitz numbers
In this section, we present our results on pruned double Hurwitz numbers. We begin by defining these objects and formulate our first main result, namely the equivalence between double Hurwitz numbers and pruned double Hurwitz numbers. This theorem expresses the electronic journal of combinatorics 22 (2015), #P00 double Hurwitz numbers as a weighted sum over pruned double Hurwitz numbers of the same genus. The rest of this section is devoted to proving this theorem.
As in [6] we define the set ℬ ( , ) of pruned branching graphs of type ( , , ) to be the subset of ℬ ( , ) consisting of all branching graphs of type ( , , ) without leaves. This leads to our main definition, which we introduce here generalizing the definition of pruned simple Hurwitz numbers in [6].
Definition 12. Let , be partitions of the same positive integer . Let be a nonnegative integer. We define the pruned double Hurwitz number to be Sometimes, we don't care about automorphisms. Thus we define the modified pruned double Hurwitz number to bê︂ 1.
By our discussion about automorphisms in Section 2, we have whenever 0 or ∞ is not fully ramified.
In fact we may express the double Hurwitz number as a weighted sum over certain modified pruned double Hurwitz numbers of smaller degree (we have to take the modified Hurwitz numbers, since removing vertices might introduce unwanted automorphisms). The idea is, that we iteratively remove all leaves of the branching graphs until none are left. To make our main result precise, we have to introduce some notation. Theorem 15. Let = ℓ( ) and let , be partitions of the same positive integer . Then we get: Moreover, by inverting the relation we see that pruned Hurwitz numbers are determined by their classical counterparts as well.
Example 17. Before we start with the proof of Theorem 15, we give some examples. The Hurwitz numbers appearing in this example were computed with GAP procedures which can be found on https://sites.google.com/site/marvinanashahn/computer-algebra.
the electronic journal of combinatorics 22 (2015), #P00 Now we may define a construction similar to the construction in the proof of Proposition 3.4 in [6]. Firstly, we introduce some new notation: Let be an ordered partition and let ⊂ {1, . . . , ℓ( )}, then we denote = ( ) ∈ . The following construction associates a pruned branching graph to a branching graph in algorithmic way. We exclude the case ℓ( ) = 1, i.e. the case of trees, since in this case our algorithm leaves a single vertex and by convention we excluded this case.
1. We remove all leaves of . That is, we remove the vertices of valency 1, all adjacent half-edges and the adjacent full-edge. Moreover, we remove all half-edges with the same label as the removed full-edge in the whole graph.
2. After that, we relabel the edges, such that the labels form a set of the form {1, .., } for some .
3. If the resulting graph˜is pruned, the process stops, if not, we start again.
When this process stops, we obtain a pruned branching graph˜of some type ( ,˜,˜) with and˜as above. We call˜the underlying pruned branching graph of . Note that we may perform this process for each face seperately. For a face , we call the resulting face˜the underlying pruned face.
We refer to Construction 18 as pruning. The resulting underlying pruned branching graph is unique.

Definition 19. Let and˜be integers with
˜and let be a rooted forest with vertices and˜components. Moreover, let the non-root vertices be bilabeled by some set and some set , i.e. each non-root vertex has two labels. Let the root-vertices be labeled by some set , such that −˜= | | = | |, | | =˜. We call a forest of type (˜, , , ). If we drop the labeling by the set , we call a forest of type (˜, , ).
Proposition 20. Let and be positive integers and fix some positive integer . Morethe electronic journal of combinatorics 22 (2015), #P00 over, let ℰ be some set of order contained in {1, . . . , }. There is a weighted bijection Faces of branching graphs on vertices with perimeter and with full-edge labels in ℰ Triples (˜, , ), such that is a pruned face of a branching graph with perimeter˜ , is forest of type (˜, , , (˜)), for some ⊂ {1, . . . , }, ⊂ ℰ, | | = | | and an ordered partition , such that While the proof of this proposition involves some intricate combinatorics, the idea is rather simple: Starting with the face of the branching graph, we associate a pruned face˜as in Construction 18. Considering the graph induced by −˜, i.e. removing the underlying pruned face, we obtain a forest . For the other direction, starting with a pruned face˜and a forest , there are several ways of reconstructing a face by gluing the forst into the pruned face.
Proof. We give an algorithm for each direction of the bijection. Let be a face of a branching graph with a total of edges, such that has perimeter with underlying pruned face˜of perimeter˜. Furthermore, let be the set of vertex-labels and the set of edge-labels not contained in˜but in . Let be the partition of the perimeters of those vertices we remove in the pruning process, such that the entries of are labeled by , i.e. the vertex labeled has perimeter . We see immediately that | | = −˜, since we remove | | · edges, where we count all full-edges twice, except the ones incident to the underlying pruned face, which we count once. We construct a forest of some type (˜, , , ).
(a) By definition each label occurs exactly˜times in˜, such that we can divide the boundary of˜in˜many segments, such that each segment is incident to an edge with a given label exactly once. By convention, each segment starts with the label 1.
We label the segments cyclically counterclockwise by 1 , . . . ,˜, where we assign 1 to the segment containing the full-edge with the smallest label in the face.
(b) Now we contract these segments in to a root vertex, one for each of the˜many components. We relabel these components by reassigning each edge label to the adjacent vertex which is further away from the root vertex. This yields the set . The root vertex is labeled by its segment, which corresponds to the set . Furthermore, each non-root vertex is by definition labeled by , thus we obtain a forest of type (˜, , , ) as above. This construction is unique.
For the other direction, we start with a tuple (˜, ), such that˜has perimeter and is a forest of type (˜, , , ). We start by labeling the segments of the boundary of as above by 1 , . . . ,˜cyclically counterclockwise, such that the segment labeled 1 contains the full-edge with the smallest label. Now, we glue the forest into the pruned face as follows: 1. We give the forest the canonical orientation. We label each edge by the label of its target-vertex corresponding to the set .
2. We introduce a partial ordering on the edges of in the following way: For two edges , ′ we define ′ , if they are contained in the same tree and lies on the unique path from the respective root vertex to ′ . We obtain a face of perimeter . One can check that both constructions are inverse to each other. The choices in step 6 are the only choices we have and thus we obtain a weighted bijection as desired.
Example 21. We use the construction in Proposition 20 in the example Figure 5. We start with a pruned face with perimeter 12. We remove vertices with labels 5 − 11 and the electronic journal of combinatorics 22 (2015), #P00 edges with labels 2, 6 − 11. The remaining labels 1, 2, 3, 5 are relabeled as 1, 2, 3, 4. We obtain a pruned face with perimeter 4, the rooted forest in Figure 5 and the partition (1, 1, 2, 1, 1, 1, 1). These objects satisfy all conditions. Proposition 22. Let = ℓ( ) and let , be partitions of the same positive integer . Then we get: Proof. The proof is similar to the proof of Proposition 3.4 in [6]. The given formula is a weighted sum over pruned branching graphs. As already seen in Construction 18, we may assign a unique pruned branching graph to each branching graph. For the other direction we apply Proposition 20 to each face iteratively. Recall that we may obtain a branching graph of type ( , , ) from a pruned branching graph of type ( , ,˜) for some , such that 1 ˜ and | | = |˜|. We can do this by choosing a decomposition = 1 ⊔ · · · ⊔ ℓ( ) , such that | | = −˜and adding vertices to the face labeled , whose perimeters correspond to , in a tree-like manner. Thus, adding ℓ( ) vertices means adding just as many edges. The desired formula may be reformulated as follows: There is weighted bijection

{︂
Branching graph of type ( , , ) Tuple ( , , ( 1 , . . . , ), ( 1 , . . . , )), such that is a pruned branching graph of type ( , ,˜) for some subset , Now we count the number of branching graphs of type ( , , ) with underlying pruned branching graph of type ( ,˜,˜). We do this by reconstructing branching graphs of type Fix a pruned branching graph of type ( , ,˜) for some ⊂ {1, . . . , ℓ( )}, such that | | = |˜|. We need to add vertices and edges as described above. Firstly, we distribute the perimeters of the vertices to the faces, that means, we choose some decomposition = 1 ⊔ · · · ⊔ , such that | | = −˜. Moreover, we distribute the edge-labels of the pruned branching graph as well as the set of edge labels, we add to face , i.e. we choose a decomposition of the 2 − 2 + ℓ( ) + ℓ( ) edge labels ( ) =˜⊔ 1 ⊔ · · · ⊔ , such that |˜| = 2 − 2 + ℓ( ) + ℓ(˜) and | | = | |. Now we may add vertices and edges as described to construct some branching graph of type ( , , ). For each branching graph constructed that way, the face contracts to some forest of type (˜, , , (˜)) as in Proposition 20. As noted in Proposition 20, each forest of type (˜, , , (˜)) corresponds to This is a generalization of the respective theorem in [6] in the sense, that for double Hurwitz numbers we obtain a weighted count over tuples of forests, where in the simple Hurwitz numbers case, each tuple is counted with weight 1. In fact, we may simplify the formula in Theorem 22, by using the following result on the number of rooted forests.
Theorem 23. Let ⊂ {1, . . . , } be a fixed set and let , be the set of rooted forests with vertices and | | components, such that the roots are labeled by .

)︂
Using this result, we see that for a fixed partition and for each degree sequence
These numbers satisfy the recursion as expected.
The idea behind this recursion is similar to the one in [6], which we aim to generalize. We start with a branching graph of type ( , , ) and remove the full-edge labeled and all half-edges with the same label. This may leave a graph that is not pruned. In that case, we apply Construction 18 and obtain a new pruned graph . We exclude the cases, where ℓ( ) 2, since our procedure is not well-defined in the case, where the graph we start with is just a cycle. Since the graph is pruned, the removed edges either form a path or look locally like the left graph in Figure 9. We can classify the possible cases for the new graph: 1. The new branching graph obtained that way is a pruned branching graph of type ( − 1, , ( ∖{ } , , )) for some subset ⊂ {1, . . . , ℓ( )}, ∈ {1, . . . , ℓ( )} and , > 0, such that + + | | = . Note, that we require for in order to be a branching graph, that its faces are homeomorphic to open disks. Thus, we need to degenerate the surface, is embedded on, as illustrated in Figure 6.
Algorithm 27. We begin this algorithm by fixing to be some pruned branching graph of type ( −1, , ( ∖{ } , , )) as in the first case. First we need to embedd on a surface of genus , such that the faces labeled ℓ( ) and ℓ( ) + 1 are joined, reversing the second step in Figure 6. We construct a pruned branching graph of type ( , , ) as follows, reversing the first step in Figure 6: 2. Choose an edge label in and attach an edge with that label to the face labeled ℓ( ) of perimeter .

Choose a vertex label in and attach a vertex of perimeter
to the other end of the edge, we attached in step 2.

5.
Choose an edge label in and attach an edge with that label to the vertex, we just attached.
8. Attach the last edge we attached to the path to the face labeled ℓ( ) + 1 of perimeter .
9. Relabel the edges of the graph without the new path by , such that the order of the edge labels is maintained.
10. Label the face obtained by joining ℓ( ) and ℓ( ) + 1 by and adjust the labels of the other faces.
The new graph obtained that way is a pruned branching graph of type ( , , ).
Algorithm 28. We begin by fixing 1 and 2 to be some pruned branching graphs of respective type ( 1 , 1 , ( 1 , )) and ( 2 , 2 , ( 2 , )) as in the second case. First, we need to embedd those graph of a surface of genus , such that the face labeled | 1 | + 1 of 1 and the face labeled | 2 | + 1 of 2 are joined, reversing the second step in Figure 7. We construct a pruned branching graph of type ( , , ) as follows, reversing the first step in Figure 7: 2. Choose an edge label in and attach an edge with that label to the face labeled | 1 | + 1 of 1 of perimeter .

Choose a vertex label in and attach a vertex of perimeter
to the other end of the edge, we attached in step 2. 8. Attach the last edge we attached to the path to the face labeled | 2 | + 1 of 2 of perimeter , joining the two graphs.
9. Relabel the edges of the graph without the new path by , such that the order of the edge labels is maintained.
10. Label the new face obtained by joining both graphs by and adjust the labels of the other faces.
The new graph obtained that way is a pruned branching graph of type ( , , ).
Algorithm 29. We begin by fixing to be some pruned branching graph of type as in the third case. We construct a pruned branching graph of type ( , , ) as follows, reversing the process in Figure 8.
2. Choose an edge label in and attach an edge with that label to the face labeled ℓ( ) − 1 of of perimeter .

Choose a vertex label in and attach a vertex of perimeter
to the other end of the edge, we attached in step 2.

5.
Choose an edge label in and attach an edge with that label to the vertex, we just attached.
In all three algorithms we have to make some choices, thus the result of each algorithm is not uniquely determined by the initial conditions. The next step in order to prove Theorem 24 is to analyze the number of choices we have in each algorithm. However, in each algorithm not every resulting graph will yield the pruned branching graph we began with, after removing the edge labeled and pruning. In the first two algorithms the graphs, where lies on the path we added will fulfil this property. In the third algorithm, we allowed the path to join itself in the last step. Thus allowing on the whole path isn't enough as illustrated in Figure 9. However, we will repair this below in the proof of Theorem 24.
We call the resulting graphs with the edge labeled on the path we attached, the relevant graphs.
1. In the first case, there are many ways to attach the first edge. There are | |! many ways to distribute the vertex labels to the path. Moreover, since we only count relevant graphs, we have | | + 1 possibilities to assign the label to some edge on the path. After assigning the label , there are − 1 many labels to assign to the − (| | + 1) − 1 edges on the path without a label, which yields a factor of ( −1)! ( −(| |+1))! . When we attach an edge to a vertex label ∈ , there are many ways to attach that edge in each step. Thus we obtain a factor of ∏︀ ∈ . Finally, no graph occurs twice in this construction, thus we proved the first statement.
2. The second case works analogously to the first one.
3. In the third case, the factors occur the same way as in the first and second case, except for the eighth and tenth step in Algorithm 29. If ̸ = in the eighth step, we have two choices to attach the last edge to the face and only one possibility in the tenth step. If = , we have only one choice in the eighth step, but two choicesin the tenth step. This would yield a factor of 2. However, the algorithm produces each graph twice by the following argument: If the path is not attached to itself, we cannot distinguish which end of the graph was attached to the face first. If the path is attached to itself, one vertex of the path is trivalent and two adjacent edges are contained in a cycle. We cannot distinguish which of those two edges was attached last. This yields a factor of 1 2 and the third statement is proved. Now, we are ready to finish the proof of Theorem 24.
Proof of Theorem 24. The three reconstructive algorithms produce all graphs of type ( , , ). We need to make sure, that each graph is obtained only once. However, we have already seen, that is not true, since the third algorithm produces graphs that contribute to the second case, as illustrated in Figure 9. However, those are exactly the graphs of the second case, where one graph is of type (0,˜,˜), such that ℓ(˜) = 2. Thus, we just exclude those cases in the second algorithm. We can also exclude those graphs with (0, ℓ(˜)) = (0, 1) sincê︂ ℋ 0 ( ,˜) = 0.
Moreover, if = in the first case, we may switch the labeling of the respective faces and the first algorithm yields the same relevant graphs. Thus, we have to adjust the count by 1 2 , if = . However, for ̸ = the first algorithm yields the same relevant graphs for graphs in ℋ −1 ( , ( ∖{ } , , )) as in ℋ −1 ( , ( ∖{ } , , )), since the construction is symmetric in and . Thus, we adjust those summands by a factor of 1 2 as well.
A similar argument accounts for the factor 1 2 in the second case and the recursion follows. Figure 9: Removing the edge labeled in the left picture corresponds to the third case in the proof of Theorem 24. However, reconstructing as in the third case, allows placing the edge labeled as in the right picture, which actually corresponds to the second case.

Polynomiality of pruned double Hurwitz numbers and connection to the symmetric group
It is well known, that double Hurwitz numbers in arbitrary genus are piecewise polynomial in the and . The first proof was given in [12]. The proof for pruned double Hurwitz numbers works analogously. We start by recalling the structure of the proof in [12]: We fix some tuple ( , , ). There are only finitely many branching graphs of that type. In each branching graph of type ( , , ) we drop the half-edges and obtain a new graph˜, which we call the skeleton of . For each type ( , , ), there are only finitely many skeletons, which may be obtained from such a branching graph. However, many branching graphs may have the same skeleton. We define ( , , ,˜) to be the number of branching graphs of type ( , , ) with skeleton˜. Thus, we may compute ℋ ( , ) as weighted sum over all skeletons, where each skeleton˜is weighted by ( , , ,˜). This is a finite sum, since all but finitely many skeletons will be weighted by 0. In [12] it was proved that ( , , ,˜) behaves piecewise polynomially in the entries of and by using Erhart theory and that each polynomial has degree 4 − 3 + ℓ( ) + ℓ( ). Thus by refining the hyperplanes, piecewise polynomiality follows for ℋ ( , ). This approach is feasible for pruned double Hurwitz numbers, since the property of a branching graph being pruned is inherent in its skeleton. Thus, ℋ ( , ) may be computed as a weighted sum over all pruned skeletons, where each skeleton˜is weighted by ( , , ,˜). The piecewise polynomiality follows analogously. The precise statement is as follows: In order to make the contributions of each skeleton more precise we introduce the notion of a reduced branching graph, which will also make the results concerning the connection to the symmetric group easier.
Definition 32. For a branching graph let be the graph obtained from by dropping all its half-edges. We call the skeleton of .
Notation 33. Let be an edge-labeled graph on a surface. We define a corner of the skeleton to be a tuple ( , , ′ , ), such that and ′ are both full-edges adjacent to and and ′ is positioned after counterclockwise. We call a corner descendant, if the label of ′ is smaller than the label of .
Definition 34. Let and be positive integers, moreover let and be ordered partitions of . We define a reduced branching graph of type ( , , ) to be a graph on an oriented surface of genus g, such that for = ℓ( ) + ℓ( ) − 2 + 2 : (v) there is at least one half-edge labeled in each descending corner.
Note that we allow loops at the vertices.
Remark 35. There is a natural bijection between branching graphs of type ( , , ) and reduced branching graphs of type ( , , ) given by pulling back an additional edge in the star graph adjacent to 0 and an unramified point and forgetting the all half-edges not labeled not labeled on the source-surface.
The contribution of each skeleton is the number of possibilities to distribute half-edges labeled to each vertex to obtain a reduced monodromy graph, such that the perimeter of the vertex labeled is and the perimeter of the face labeled is . We compute the standard and pruned polynomials in one example.
Example 36. We compute the polynomials in genus 0 for the double Hurwitz numbers ℋ 0 (( , ), ( , )) and their pruned counterparts ℋ 0 (( , ), ( , )). In this simple case, we can read the contribution directly from the graph without using the procedure of the proof. All possible skeletons are illustrated in Figure 10 (in what follows, we enumerate the graphs from the top left to bottom right along the rows). Only the first two are pruned. We compute the polynomial for the chamber < , < .
the electronic journal of combinatorics 22 (2015), #P00 1. The first two skeletons each contribute a factor of : We need to attach half-edges labeled to the vertex labeled 1 and half-edges labeled to the vertex labeled 2, such that the face labeled 1 has perimeter and the face labeled 2 has perimeter . Since < , for any ∈ {1, . . . , }, we can attach half-edges labeled to the vertex labeled 1, such that these half-edges are contained in the face labeled 1. This determines the entire graph, thus we have |{1, . . . , }| choices.
2. The third and fourth graph each contribute a factor of − .
3. The fifth and sixth graph contribute a factor of 0. In Section 2 we have explained the connection between Hurwitz numbers and branching graphs and the connection between Hurwitz numbers and factorizations in the symmetric group. The proof of Theorem 6 yields the following algorithm, which yields the connection between branching graphs and factorizations in the symmetric group. In [14], a similar algorithm is given, which for a given Hurwitz galaxy yields a representation in the symmetric group. However, this algorithm produces the products of permutations = . . . 1 1 from which we can recursively deduce ( 1 , 1 , . . . , , 2 ). Our algorithm produces ( 1 , 1 , . . . , , 2 ) as in Theorem 6 directly and is a direct consequence of the mondromy representation of a branched holomorphic covering.
the electronic journal of combinatorics 22 (2015), #P00 Definition 37. Let be a reduced branching graph of type ( , , ) we call the conjugacy class of the tuple ( 1 , 1 , . . . , , 2 ), that is produced by the algorithm below, the monodromy representation of .
The notion of a monodromy representation of a branched covering in the literature is closely related to the notion defined above. Namely, one can think of a monodromy representation of a cover as a choice of a tuple ( 1 , 1 , . . . , , 2 ) as in Theorem 6. To be more precise: Let = { 1 , 2 , 1 , . . . , } be the set of branch points on P 1 (C), then the tuple ( 1 , 1 , . . . , , 2 ) defines a group homomorphism Φ : 1 (P 1 (C∖ ) → . We will see in Proposition 40 that the monodromy representations of a branched covering and of its corresponding branching graph coincide.
Algorithm 38. Let be a reduced branching graph of type ( , , ).  4. Define ( 2 ) −1 to be the permutation whose -th cycle is given by the cyclic numbering of labels of half-edges in the -th face and label the -th cycle by .
This gives a tuple ( 1 , 1 , . . . , , 2 ) as in Theorem 6. Note, that we have a choice in the first step of Algorithm 38, namely we didn't specify where the enumeration starts. However, this just corresponds to conjugations of the resulting monodromy representation, thus the resulting conjugacy class of the algorithm is well-defined. Proposition 40. The monodromy representation of a branched covering and of its corresponding branching graph coincide.
The proof is similar to the discussion in Section 4 in [14]. Now, we will pick up our discussion about automorphisms in Section 2. One can check, that two branching graphs and ′ are isomorphic, if their corresponding monodromy representations coincide. On the other hand, the conjugation of a tuple in the monodromy representation yields another isomorphic branching graph by relabeling. That means isomorphims between branching graphs correspond to conjugations of the results of Algorithm 38. It follows, that automorphisms correspond to conjugations that preserve the result of Algorithm 38.
the electronic journal of combinatorics 22 (2015), #P00 However, due to transitvity and the fact that we labeled the disjoint cycles of 1 and 2 , it follows that only tuples, where 1 and 2 are -cycles may be invariant under non-trivial conjugations.
We finish this section by giving a classification of pruned Hurwitz numbers in terms of factorizations in the symmetric group, which is an immediate consequence from Algorithm 38.
Proof. To begin with, we prove that for each pruned branching graph of type ( , , ), Algorithm 38 produces such a representation. The only condition to check is the last one, but this is immediate, because each cycle 1 corresponds to a vertex . This vertex is not a leaf, because the branching graph we began with is pruned. Thus, there are two edge and ′ adjacent to . However, these edges correspond to two transpositions and ′ , that by construction fulfil the last condition. The other direction follows similarly from the fact, that the monodromy representation of a branching graph is the same as the monodromy representation of the corresponding cover. We excluded = 1, due to the fact, that we assume the graph consisting of only one loop and one vertex to be pruned. his help with Figure 6. The author gratefully acknowledges partial support by DFG SFB-TRR 195 "Symbolic tool in mathematics and their applications", project A 14 "Random matrices and Hurwitz numbers" (INST 248/238-1).