#### Abstract

We give a bijective proof for a relation between unicellular, bicellular, and tricellular maps. These maps represent cell complexes of orientable surfaces having one, two, or three boundary components. The relation can formally be obtained using matrix theory (Dyson, 1949) employing the Schwinger-Dyson equation (Schwinger, 1951). In this paper we present a bijective proof of the corresponding coefficient equation. Our result is a bijection that transforms a unicellular map of genus into unicellular, bicellular or tricellular maps of strictly lower genera. The bijection employs edge cutting, edge contraction, and edge deletion.

#### 1. Introduction

A -cellular map is a fatgraph having boundary components that are connected as a combinatorial graph. It can be interpreted as a cell complex whose geometric realization is a surface and as such encodes the invariants of the latter, as genus and orientability.

A -cellular map is called unicellular and similarly -cellular and -cellular maps are referred to as bicellular and tricellular maps, respectively. Unicellular maps are also known as fatgraphs [13], having a unique boundary component.

Unicellular maps were of central importance in a seminal paper of Harer and Zagier [4], who computed the virtual Euler characteristics of the Moduli space of curves. The virtual Euler characteristics were independently derived by Penner [5] by means of orthogonal polynomials. The key computation in [4] is that of the generating function of unicellular maps of genus having edges, . Their numbers, , satisfy the recursion: Another genus recursion, obtained by different means, namely, by splicing vertices, while keeping the number of edges constant, was derived in [6]. In the context of matrix theory [7, 8], the Schwinger-Dyson equation implies a relation of the generating functions of unicellular, , and bicellular maps, . This relation can also be obtained using representation theoretic framework [9] and is given by Recently, in [10] the authors presented a bijective proof of the corresponding coefficient equation of (2): which revealed a simple construction mechanism. The bijective proof can, for instance, be applied, to significantly speed up the folding of RNA interaction structures [11, 12].

This paper presents the bijective proof of the analogue of (3), relating to unicellular bicellular and tricellular maps. Formally, this relation be obtained via matrix theory and reads where is expressed in terms of the numbers of unicellular and bicellular maps. For the proof it is important to identify a suitable partition of the set of unicellular maps of genus with edges.

It is worth mentioning that matrix theory does not provide insight into a recursion involving quadricellular maps. In fact it seems unlikely that such a relation can be derived using the formal framework. The bijective proof presented here is however rather intuitive, once the correct partitioning is identified. We believe that it is possible to prove similar relations for maps with more than three boundary components.

#### 2. Basic Definitions

Let denote the permutation group over elements.

Definition 1. Let be positive integers. A -cellular map is a triple , where is a set of cardinality , is a fixed-point free involution and are cycles such that . The elements of are called half-edges, and the cycles of are called edges. The cycles of the permutation are the vertices , . The length of is its degree. The cycle is called the th face.

The associated combinatorial graph of a -cellular map is the graph whose edges and vertices are the cycles of and . We can regard a -edge as a ribbon whose two sides are labeled by the half-edges as follows: each side of the ribbon represents one half-edge, we decide which half-edge corresponds to which side of the ribbon by the convention that, if a half-edge belongs to a cycle of and a certain of , and then is the right-hand side of the ribbon corresponding to , when entering .

We draw the graph in such a way that, around each vertex , the counterclockwise ordering of the half-edges belonging to the cycle is given by that cycle. This ordering of half-edges enriches the combinatorial graph to a ribbon graph or fat graph . Clearly, a fat graph with its boundary components is tantamount to the -cellular map . is interpreted as the cycle of half-edges visited when making the tour of the graph, keeping the graph on its left.

For the orbits of under the action of the subgroup of generated by , , and are called the connected components of the -cellular map. If this action is transitive, we say that the associated graph and fat graph of the -cellular map are connected. Topologically, is a deformation retract of the surface with boundary .

Definition 2. A planted -cellular map is a -cellular map in which each contains a distinguished half-edge , such that is a -cycle. is called the plant of the face and -cycles, except that the plants are called vertices.

In the following, we refer to edges not incident to plants as edges. Let denote the set of planted -cellular maps that contain edges.

In planted maps we will label the half-edges of such that , that is, Given we define the linear order on for each face via Let furthermore denote the set consisting of the half-edges in one of these faces , . In particular, is the set of half-edges contained in the face .

There is a natural equivalence relation over half-edges, and in particular, . If , then is called a one-sided edge and is called a two-sided edge, otherwise.

For each vertex , let denote the first half-edge via which enters . This gives a canonical way of writing the cycle, starting at , namely, . In particular, the vertex containing the half-edge , , is the “first” vertex.

#### 3. The Partition

Let denote the set of planted, unicellular maps of genus , having edges. In particular, let denote the unicellular map of genus zero, containing no edge. This map contains only one edge, the plant, and one additional vertex.

Let with face

Then where . Thus and . In the following we will identify a partition of that shall facilitate our main result (Theorem 11).

We consider for the four half edges , , and . Clearly, , whence . Furthermore, by construction, see also Figure 1. Accordingly, there are the following two scenarios: The case belongs to scenario , which then reduces to This generates the bipartition of ,

We next refine : for , we consider the cycle and we use to split into three branches , , and , where see Figure 1.

Suppose the restriction is a well-defined fixed-point free involution, and then we call closed. Similarly, the sets and , are called closed, if and are fixed-point free involutions.

Let denote the subset of elements in which no is closed and let denote its complement. Then We refine further by considering how many branches the cycle contains: (i): the set of elements such that exactly two are empty,(ii): the set of elements such that exactly one is empty,(iii): the complement of and , that is, the set of such that no is empty.

Thus

We refine a bit more; for this purpose,(i)let be the subset of elements where ,(ii)let be the subset of -elements where , and ,(iii)let be the complement of and , that is, subset of elements where

Accordingly,

Furthermore we present : where (i) denotes the subset of elements with ,(ii) denotes the subset of elements with .

We write as where (i) is the subset of elements with ,(ii) is the subset of elements with ,(iii) is the subset of elements with and .

#### 4. Some Lemmas

In this section we state three procedures that are employed repeatedly in our bijection. They are subsequently referred to as “cutting,” “contracting,” and “deleting.” These procedures constitute the key three operations that, applied in various contexts, eventually facilitate the bijection.

Lemma 3 (cutting). Suppose we are given a planted, unicellular map with Then can be mapped to a planted, -cellular map, , with the three faces , and via where has the following inverse:

Proof. By assumption we have whence the face of can be written as in (21). We use and , which are given by (14), then concatenate the sequence of half-edges of , , and to form and relabel the cycles as in (23). This produces the plants , , and . Since , is a -cellular map and accordingly is welldefined, see Figure 2.
We next construct an explicit inverse of . Suppose we are given a -cellular map , in which the are as in (26). Then we concatenate the sequences of half-edges of the three -cycles and relabel as in (21); that is, , , and . We derive, by construction, Accordingly, is a unicellular map of genus with property .

Lemma 4 (contraction). Suppose has a one-sided edge , , such that and are incident to two different vertices . Relabeling the two half-edges we can write the face:
Here either or , or and either or . Then corresponds to a unicellular map together with two distinguished half-edges via mapping where , , and , if ; , if and ; , , if and ; and finally , if .
Furthermore, has the property .

Proof. is by construction unicellular and retains the genus of .

We describe the contraction in Figure 3.

Lemma 5 (deletion). Given a unicellular map with face where or , or and or .
Then corresponds to a unicellular map together with two half-edges and , where , via the mapping where , and can be reversed by mapping a unicellular map , together with two arbitrary half-edges and () as follows:

Proof. By construction, , is a fixed-point free involution and has cardinality , whence is unicellular. Euler characteristic implies that the genus of is . Moreover, we set see Figure 4.
Given a unicellular map , there are ways to choose such that . We now select two half-edges such that and insert the pairs of half-edges , into the face . This produces the face , with , , and , and . Consequently we have We then relabel as in (31). Since is a fixed-point free involution and is a set of cardinality , is a unicellular map with property . Euler characteristic implies has genus . By construction, we have .

#### 5. The Main Theorem

In this section we state some auxiliary bijections and our main result. We furthermore give in Figure 5 a modular description of how our bijection works.

We call a planted -cellular map, whose combinatorial graph is connected, a planted, bicellular map. Let denote the set of planted, bicellular maps of genus with -edges.

Let denote the subset of in which only a single , is closed and let denote the set of elements in which all , are closed, that is, .

Lemma 6. We have the bijections

We prove Lemma 6 in Section 6.

Lemma 7. We have the four bijections:

We prove Lemma 7 in Section 6.

Lemma 8. We have the three bijections:

We prove Lemma 8 in Section 6.

Let denote the set of planted, tricellular maps of genus with edges.

Proposition 9. There is a bijection:

We prove Proposition 9 in Section 6.

Proposition 10. There is a bijection

We prove Proposition 9 in Section 6. In Figure 5 we give an overview of how the above bijections are applied.

For a set we denote its cardinality by .

Theorem 11. Consider where with

#### 6. Proofs

##### 6.1. Proof of Lemma 6

Proof. Consider the following.
Claim  1. The mapping: is a bijection. We first prove that is welldefined. For a planted unicellular map with face we employ the mapping of the Cutting-Lemma (Lemma 3) in order to decompose into a planted -cellular map, , where where is obtained by concatenating the sequence of half-edges of , , and .
For any , is closed. Since , the restriction is a fixed-point free involution. Accordingly, is a planted unicellular map.
Since is closed and , is given in (48), the restriction is a well-defined fixed-point free involution. Furthermore, since neither nor are closed, and are not closed either. Therefore is a planted bicellular map with the plants and .
Let and .
Suppose , , and have , and vertices, respectively. Then and , whence Since the edges incident to plants and plants do not contribute to the number of edges and vertices, we have , . As a result has genus , where , whence is well-defined.
We next show that is injective. In order to apply the mapping of the Cutting-Lemma, we introduce where are given by (48), , and .
For any where , and , we apply . This generates the -cellular map . Since has edges and has edges, and the process generates the edges and , we have . We can now apply of Lemma 3, which induces the mapping . Lemma 3 now implies furthermore whence the mapping is injective.
It thus remains to prove that is surjective. This follows again from close inspection of the proof of the Lemma 3, which implies Therefore, is surjective and Claim  1 is completed.
Analogously we prove that and are injective.
Claim  2. The mapping: with and is a bijection.
We first show that is well-defined. As in the proof of Claim  1, we employ the Cutting-Lemma which produces a -cellular map with the boundary components .
For any , each of the is closed. Thus the restrictions , for , are welldefined and fixed-point free involutions. As a result, , , and are unicellular maps, respectively.
Let
Suppose that , , , and have , , , and vertices, respectively. Then Furthermore, we have After applying the Cutting-Lemma, and become plants, similarly and become edges incident to plants. Thus, we have and and accordingly obtain Consequently, has genus , where , and is well defined.
We next prove is injective. We establish this as in Claim  1, introducing where is given in (48), , and . Analogously, of Lemma 3 induces the mapping and whence the mapping is injective.
Subjectivity of is implied by the Cutting-Lemma which guarantees whence Claim 2 and the proof of the lemma are complete.

##### 6.2. Proof of Proposition 9

Proof. We prove that the mapping is a bijection. As for well definedness, suppose where We use mapping of the Cutting-Lemma and derive the planted -cellular map, , where Here is obtained by concatenating the sequence of half-edges contained in , and .
For none of the , is closed, whence the associated combinatorial graph of is connected. Accordingly, is a planted tricellular map with plants , , and . Euler's characteristic formula implies has genus and edges, whence is well defined. Injectivity and surjectivity of are implied by the Cutting Lemma.

##### 6.3. Proof of Lemma 7

Proof. Consider the following.
Claim  1. The mapping: is a bijection.
The contraction lemma implies that is well defined. Injectivity of follows by considering the mapping induced by the mapping of Lemma 4, where . Lemma 4 guarantees , whence is injective. Surjectivity of is a consequence of , implied by Lemma 4; see Figure 6.
The proof that is a bijection for follows analogously; see Figure 7.

##### 6.4. The Proof of Lemma 8

Proof. Consider the following.
Claim  1. The mapping: is a bijection.
We first prove that is well defined. Consider together with two one-sided edges, , , such that and are incident to , and are incident to and .
We then apply Lemma 4 to a together with the one-side edge . We iterate applying Lemma 4 w.r.t. the edge . By definition of Lemma 4 this generates the unicellular map of genus having edges with distinguished four half-edges , , and .
Since Lemma 4 preserves genus, has genus and edges, whence is well defined.
We next prove is injective. Suppose we have a unicellular map with four distinguished half-edges , , , and . We observe that the mapping constructed in Lemma 4 allows us to obtain a mapping such that whence injectivity.
Surjectivity follows by computing .
The proof that are bijections is analogous; see Figure 8.

##### 6.5. The Proof of Proposition 10

Proof. Proposition 10 follows directly from Lemma 5.

##### 6.6. The Proof of Theorem 11

Proof. According to Lemmas 6, 7, and 8 and Propositions 9 and 10, we have (i), for ,(ii), for ,(iii), for ,(iv),(v).
According to (12), (15), and (16) we have Furthermore, according to (18), (19), and (20), we have which establishes (43).

#### Acknowledgments

The authors wish to thank Fenix W. D. Huang and Thomas J. X. Li for discussions. This work is funded by the Future and Emerging Technologies (FET) programme of the European Commission within the Seventh Framework Programme (FP7), under the FET-Proactive Grant Agreement TOPDRIM, FP7-ICT-318121.