Simple formulas for constellations and bipartite maps with prescribed degrees
We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, -angulations), and constellations. These formulas are the fastest known way of computing these numbers.
Our work is a natural extension of previous works on integrable hierarchies (2-Toda and KP), namely the Pandharipande recursion for Hurwitz numbers (proven by Okounkov and simplified by Dubrovin–Yang–Zagier), as well as formulas for several models of maps (Goulden–Jackson, Carrell–Chapuy, Kazarian–Zograf). As for those formulas, a bijective interpretation is still to be found. We also include a formula for monotone simple Hurwitz numbers derived in the same fashion.
These formulas also play a key role in subsequent work of the author with T. Budzinski establishing the hyperbolic local limit of random bipartite maps of large genus.
Keywords: maps, Hurwitz numbers, Toda hierarchy, constellations
A map is a combinatorial object describing the embedding up to homeomorphism of a multigraph on a compact oriented surface. A bipartite map is a map with black and white vertices, each edge having a black end and a white end. Constellations are generalizations of bipartite maps with more colors (see Section 2 for precise definitions).
Map enumeration has been an important research topic for many years now, going back to Tutte [Tutte] with planar maps. He used analytic techniques on generating functions, and later on, Schaeffer enumerated planar maps bijectively [theseSch], with many generalizations (see for instance [BDG, BF, AP, CMS, Lep]). The enumeration of maps was extended to other models : for instance, asymptotic formulas were obtained by Bender and Canfield [BC] for maps of higher genus, by Gao [Gao] for maps with prescribed degrees, and Chapuy [Constellations] for constellations. Another way to count maps is to see them as factorizations of permutations and to use algebraic properties of . In particular, maps fit in the more general context of weighted Hurwitz numbers (see e.g. [ACEH]). Their generating functions satisfy integrable hierarchies of PDEs that arose from mathematical physics, namely the KP and 2-Toda hierarchies (a good introduction can be found in [Solitons]).
The first numbers that were studied from the point of view of integrable hierarchies were Hurwitz numbers, that enumerate ramified coverings of the sphere. Pandharipande conjectured a recurrence formula for those numbers [Pandharipande], which was proven by Okounkov [Ok] and later simplified by Dubrovin, Yang and Zagier [Zagier]. Later, recurrence formulas for maps were found, starting with Goulden and Jackson for triangulations [GJ]. They were followed by Carrell and Chapuy for general maps [CC], and Kazarian and Zograf for bipartite maps [KZ]. All these works start from the fact that an underlying generating function is a "tau function" of an integrable hierarchy, and then use ad-hoc techniques to obtain explicit recurrence formulas. The generality of this second step is not well understood. The approach developped in [GJ, CC, KZ] does not generalize to constellations, and neither to controlling face degrees (except for the particular minimal case of triangulations [GJ]). On the other hand, in [Ok, Zagier], formulas are derived only for Hurwitz numbers unramified at and (which corresponds to maps without control over the degrees).
Contributions of this article : We manage to combine these two approaches in the context of maps, and we derive recurrence formulas for bipartite maps with prescribed degrees, allowing us in particular to derive a formula for bipartite -angulations. We also find recurrence formulas for constellations.
These formulas are, up to our knowledge, the simplest and fastest way to calculate those numbers (in all models, it takes arithmetic operations to calculate the coefficient for edges and genus , see Remark 2.1).
In addition to the computational aspect, such recurrence formulas are the only tool we know of in the study of asymptotic properties of large genus maps: the Goulden–Jackson formula played a key role in the recent proof [BudzLouf] of the Benjamini–Curien conjecture [PSHT] of the convergence of random high genus triangulations towards a random hyperbolic map. Similarly, the results of this paper are necessary in the study of random high genus bipartite -angulations in an article in preparation by T. Budzinski and the author.
Structure of the paper : In Section 2, we will give precise definitions and state our main results. The rest of the paper presents the main steps of the proof. The first part of the proof is common to all models : we introduce the "tau function" , a certain generating function for constellations. This function, along with some auxilliary functions , classically satisfies a set of differential equations called the -Toda hierarchy. Our first contribution, inspired by [Ok], is to link to the and derive an equation involving only (Proposition 3.3). This will be presented in Section 3. From this equation, specialized to the model we wish for (bipartite maps or constellations), we perform a few combinatorial operations (that are specific to the model, similarly as in [CC, GJ, KZ]) to obtain our formulas. We will present this in details for bipartite maps in Section 4, and we briefly mention the case of constellations. In Section 5, we will present additional models, especially one-faced constellations, and in Section 6 we will derive a similar formula for (simple, unramified) monotone Hurwitz numbers.
2 Definitions and main results
A map is the data of a connected multigraph (multiple edges and loops are allowed) (called the underlying graph) embedded in a compact oriented surface , such that is homeomorphic to a collection of disks (this implies in particular that is connected). The connected components of are called the faces. The genus of is the genus of (the number of "handles" in ). is defined up to orientation-preserving homeomorphism. A bipartite map is a map with two types of vertices (black or white), such that each edge connects two vertices of different colors. A bipartite map is said to be rooted if a particular edge is distinguished.
An -constellation is a particular kind of map with two kinds of vertices : colored vertices, carrying a "color" between and , and star vertices. Each edge connects a star vertex to a colored vertex. A star vertex has degree , and its neighbors have color ,,…, in the clockwise cyclic order. A constellation is said to be rooted if a particular star vertex is distinguished. A constellation with star vertices is said to be labeled if each star vertex carries a different label between and . Since rooting kills all possible automorphisms, there is a -to- correspondence between labeled and rooted constellations with star vertices. From now on, we will only consider rooted objects unless stated otherwise.
Some basic, well-known, properties of maps and constellations will be useful later.
Labeled (non-necessarily connected) -constellations with star vertices are in bijection with -uples of permutations of such that . The permutation represents the vertices of color : each vertex is a cycle of , and the elements of the cycle represent the neighboring star vertices, in that cyclic order. The permutation encodes the faces, see Figure 1 for an example. Bipartite maps are in bijection with -constellations, since each star vertex and its two adjacent edges can be merged into a single edge connecting a black and a white vertex.
Our main results are the following theorems :
The number of bipartite maps of genus with faces of degree (for ) satisfies :
where , , , and (the ’s count edges, the ’s count vertices, in accordance with the Euler formula).
The numbers of -constellations of genus with star vertices satisfy the following recurrence formula :
Theorem 1 has an immediate corollary, i.e. a recurrence formula for bipartite -angulations:
The number of bipartite -angulations of genus with faces satisfies the following recurrence formula :
Theorem 1 allows to compute the number of maps with prescribed degrees way faster than the usual Tutte-Lehman-Walsh approach [LW, BC, Gao] or the topological recursion (see e.g. [Eynard]), especially for large genus (because these methods require counting maps with up to boundaries to enumerate maps of genus ). It can also be specialized to maps with bounded face degrees (contrarily to the Tutte equation). Note that, in order to compute the coefficients recursively, a term from the RHS has to be move to the LHS.
The coefficients in our recurrence formulas have a combinatorial flavor. It is a natural question to ask for a bijective proof of these formulas. However, the bijective interpretation of formulas derived from the KP/2-Toda hierarchies is still a widely open question, as bijections have only been found for certain formulas, in the particular cases of one-faced [CFF] and planar maps [Louf].
3 Constellations and the Toda hierarchy
3.1 The semi-infinite wedge space
We give some definitions, mostly following the notations of the appendix in [Ok2] :
A Maya diagram is a decoration of with a particle or an antiparticle at each position, such that for some there are only particles at positions and only antiparticles at positions . The semi infinite wedge space is the vector space whose orthonormal basis elements are the Maya diagrams.
For any , we define the fermion operators and . For each Maya diagram m, we set:
where is the number of particles of m is positions (it is finite by definition of a Maya diagram). Also, is the same as m except there is a particle in position , and is the same as m except there is an antiparticle in position . Note that and are dual operators.
We can now define the boson operators : for all , let
Finally, the two vertex operators are
We will now define diagonal operators over and relate Maya diagrams to partitions.
We define the normally ordered products
Note that, for a Maya diagram m
The charge operator is:
The eigenvectors of are the Maya diagrams. The eigenvalue of a Maya diagram m is the number of particles in positive position minus the number of antiparticles in negative position. We call this number the charge of m. We introduce the translation operator : for any m, has a particle in position if and only if m has a particle in position . Note that if the charge of m is , the charge of is , and that .
There is a bijection between Maya diagrams of charge and partitions, as depicted in Figure 2 (in position , a down-step corresponds to a particle, an up-step corresponds to an antiparticle). Thus, any Maya diagram m can be encoded by its charge and a partition (that corresponds to the Maya diagram ).
We will use the braket notation, and denote the Maya diagram corresponding to the empty partition by , and set . We will also set to be the Maya diagram of charge corresponding to the partition .
Finally, we define the energy operator
In particular, , where is the number of boxes in .
3.2 Generating functions as tau functions
Fix integers , and , fix two partitions of . Let be the number of -uples of permutations of such that and has cycles, and have respective cycle types and . The enumerate (labeled, non-necessarily connected) constellations, in accordance with Proposition 1. Let be the associated generating function (that implicitly depends on ):
It is a classical result (under different forms and variants, see for instance [GJ, Ok]) that the function can be expressed in terms of elements and operators of :
Lemma 1 (Classical).
First, we have
where the are the contents of the partition (see Figure 2). It can be shown using the Jacobi-Trudi rule (see e.g. [Ok2]) that
where the sum spans over all partitions.
Thus the RHS of (3.1) can be rewritten as :
This expression (the "content product form") is equal to (see e.g. [GJ], Theorem 3.1).
We introduce the auxiliary functions , for :
We have . The previous lemma, along with classical considerations (see for instance Section 2.6 in [Ok]), imply that the satisfy an infinite set of equations, the -Toda hierarchy. In particular, the following equation holds :
So far, the content presented was classical. Our first main contribution is to transform the previous equation into an equation implying only.
3.3 The master equation
In this section we derive the following general equation:
The general generating function of connected constellations satisfies:
with and .
We only included the arguments of when they differ from .Note that there is no ambiguity in writing something like , since .
This formula will be the starting point for all the particular cases we will consider in the next section: for each model, we will apply a particular specialization of the variables, then interpret combinatorially the operator (depending on the model), and finally the extraction of coefficients will give us the relevant formulas.
We first need to relate the auxilliary functions and to the generating function .
We will describe how , and behave under the action of the shift operator, then using the operator form (3.1) of we will derive the result.
It is easily verified that the opertors , and commute with and . We also have .
By a careful change of indices,
Since , we have
The idea of expressing in terms of by calculating is inspired by the calculation performed in [Ok], Section 2.7.
We can now prove Proposition 2.
4 Proof of the main formulas
In the following subsections, we will specialize some of the variables to fit the cases we care about. To avoid tedious notations, and as there is no risk of ambiguity, the specialization of the function will still be called .
4.1 Bipartite maps
In this section, we want to count bipartite maps while controlling the degrees of the faces. Thus, we will consider the case , and specialize by setting and .
Let be the number of bipartite maps of genus with faces of degree , and be the ordinary generating function of connected rooted bipartite maps, defined as
with and (Euler formula).
Equation (3.3) can be rewritten in terms of only:
In this section, is the (exponential) generating function of labeled bipartite maps, and as mentioned in Definition 1, there is a correspondence between labeled and rooted bipartite maps. Thus
We will now express in terms of . The specialization implies that only the terms form the original function survived, and thus in this case
Finally, is the exponential generating function of labeled bipartite maps with a marked digon here is a -to- correspondence between labeled and unlabeled-unrooted bipartite maps with a marked digon and edges. A marked digon can be contracted into a root edge (see Figure 3) except when the bipartite map is just one edge, thus (the factor comes from the fact that we lose an edge when we contract the digon, and the term is the case where we cannot contract the digon). ∎
We are finally ready to prove Theorem 1.
In this section, we will count constellations without controlling the degrees of the faces. For that, we will specialize by taking , , and for all . The variable counts the number of colored vertices plus the number of faces, or equivalently, by Euler’s formula, the genus.
Proof of Theorem 2.
After the specialization, becomes by the same argument as in the proof of Theorem 1. If we take to be the (ordinary) generating function of connected constellations, we have, as before, . Equation (3.3) becomes
This time, we cannot track the degrees of the faces, as in general the combinatorial operation of contracting an -gon might disconnect the map, and the formula gets messy. However, if we restrict to only one face we can perform this operation to recover a nice formula (see Section 5).
5 Additional results
5.1 One-faced constellations
In this section, we will derive a recurrence formula for constellations with one face. In the case of bipartite maps, the formula is just a particular case of (2.1), but for , it cannot be derived from (1) directly. One-faced constellations were first enumerated in [PS]: an exact formula given the degree distribution of each colored vertex is provided. While the following formula does not give control over the degrees of the vertices, it is much quicker to calculate the "global" (i.e. controlling only the genus and the number of vertices) number of one-faced constellations for (for , i.e. bipartite maps, a nice formula for one-faced bipartite maps can be found in [Adrianov]).
Let be the number of one-faced -constellations of genus with star vertices. Also, let be the number of one-faced -constellations of genus with star vertices and distinguished colored vertices, i.e. . We have the following recurrence formula:
This formula reminds of the formula for one-faced maps proven bijectively by Chapuy in [trisections]. Indeed, it allows to calculate the number of one-faced maps of genus in terms of number of maps of lower genus with the same number of edges and some distinguished vertices. The difference, although, is that in Chapuy’s formula there are an odd number of distinguished vertices, whereas in (5.1) there are an even number of distinguished vertices.
Nevertheless, there might be a connection as those formulas arise in the same algebraic context. Our formula is obtained via the 2-Toda hierarchy, whereas Chapuy’s is an intermediate step to prove the Harer-Zagier recurrence formula (see [CFF]), which is itself a special case of a formula obtained via the KP hierarchy: the Carrell-Chapuy recurrence formula [CC].
To prove (5.1), we will take and specialize as follows: fix an integer , and set , for all , as well as . Set also for all , and extract the coefficient of . is now simply a polynomial in . It counts labeled one-faced constellations. Let be the associated polynomial for rooted objects, the classical correspondence between labeled and rooted objects yields . As before, there is a "marked -gon", and we need to interpret this combinatorially:
After the specialization, the LHS of (3.3) becomes
The only terms of that survive the specialization are those in . The first factor comes from the derivation by . We are counting constellations with a marked -gon, as in the proof of Theorem 1, it kills all symmetries, thus there is a -to- correspondence if we forget the labels.
We will work with permutations to make things easier. We are now counting -uples of permutations of with the constraint . Since we work with unlabeled constellations, we can arbitrarily fix some things:
In cycle products, is written .
The image of by is .
Such -uples of permutations are in bijection with connected unlabeled -constellations with one face of degree and one face of degree .
We can describe the "-gon contraction" on the permutations. To we will associate a -uple of permutations of :
To , we associate
For , to we associate the permutation where in the cycle product we just deleted the element (see Figure 4)
To we associate
This exactly describes a rooted -constellation with one face of degree . To go back, one needs to remember, for , what was the preimage of in (including possibly itself). There are possible choices for each , thus after the specialization, becomes .
Finally, knowing that completes the proof.
A simple calculation in the right-hand side finishes the proof:
Proof of Theorem 3.
In the RHS, we have a product of two terms. Since has no constant coefficient in the ’s, after specialization we get the coefficient of of (which is just , corresponding to the constellation with only one star vertex) times the coefficient of in
Again, since , we can do the same coefficient extraction as in the proof of Theorem 1, and obtain the result. ∎
5.2 Controlling more parameters
In each of the previous cases, we specialized a lot of variables to obtain formulas for "global" coefficients. Starting over from (3.3) without specializing some of the variables, one is able to obtain (slightly more complicated) formulas for more fine-grained coefficients. As an example, we can calculate the number of -constellations of genus , with star vertices and faces:
where the sum is over , , , and .
Even though the summation is complicated, (5.2) allows to compute all the coefficients from the initial condition iff and , and otherwise.
However, it does not restrict to a formula for one-faced constellations.
We can also find formulas for other models, with other specializations. Relevant models include bipartite maps (with prescribed face degrees), one-faced constellations, or (general) constellations, with control over the number of vertices of each color. We can also obtain a formula for triangulations (by specializing , , ), but it is more complicated (and less "combinatorial") than the Goulden–Jackson formula [GJ]. The reader is encouraged to play with (3.3) to find other nice formulas.
5.3 Univariate generating series
A relevant corollary of our results is that the formulas we obtain allow to compute the univariate generating series of some given models of maps ( angulations counted by faces, constellations counted by star vertices, etc.). To illustrate this fact, fix an integer and let be the generating series of genus bipartite -angulations:
Our formula gives an algorithm to compute every for , given . Indeed, take , Corollary 1 rewrites
where , and is a polynomial in its variables and their (first and second) derivatives. It is well known (see for instance [BDG])) that
with the change of variable
Note that we have a "" in the expression of because we do not count the "empty map".
Assuming we know for , this gives a linear, second order ODE in (with respect to the variable ). Since all the ’s are rational in (see for instance [ChapuyFang]), all the coefficients of the equation are themselves rational, and the solutions can be computed explicitly. The initial conditions are given by the two following facts : and is the number of unicellular bipartite maps of genus with edges, that can for instance be computed using Theorem 3.
6 Monotone Hurwitz numbers
In this section, we derive a recurrence formula for monotone Hurwitz numbers, in a similar fashion as in previous sections. These numbers, which appear in the calculation of the HCIZ integral, were introduced in [monotone].
For two transpositions of , we say that if . The double monotone Hurwitz number is times the number of tuples of permutations of such that:
where is the number of parts of
is an increasing sequence of transpositions
(resp. has cycle type (resp. )
the permutations act transitively on
The simple monotone Hurwitz numbers are defined as .
We will set to be the same numbers without the transitivity condition, and introduce
with such that . is the generating function of the .
As before, it can be shown (see for instance [GPH]) that
with and .
Similarly as with constellations, in general we cannot even track the cycle type of , although, from the specialization for all we can obtain a recurrence formula for the unramified monotone Hurwitz numbers :
In this paper, the number are defined with a scaling factor of to make the formula simpler, this is a different convention as in [monotone].
In genus , we have . There is an elementary proof of this result, we leave it as an exercise to the reader.
The author wishes to thank Guillaume Chapuy for suggesting the problem and for useful discussions.