Enumeration of non-oriented maps via integrability

In this note, we examine how the BKP structure of the generating series of several models of maps on non-oriented surfaces can be used to obtain explicit and/or efficient recurrence formulas for their enumeration according to the genus and size parameters. Using techniques already known in the orientable case (elimination of variables via Virasoro constraints or Tutte equations), we naturally obtain recurrence formulas with non-polynomial coefficients. This non-polynomiality reflects the presence of shifts of the charge parameter in the BKP equation. Nevertheless, we show that it is possible to obtain non-shifted versions, meaning pure ODEs for the associated generating functions, from which recurrence relations with polynomial coefficients can be extracted. We treat the cases of triangulations, general maps, and bipartite maps. These recurrences with polynomial coefficients are conceptually interesting but bigger to write than those with non-polynomial coefficients. However they are relatively nice-looking in the case of one-face maps. In particular we show that Ledoux's recurrence for non-oriented one-face maps can be recovered in this way, and we obtain the analogous statement for the (bivariate) bipartite case.


INTRODUCTION AND MAIN RESULTS
In this note, we are interested in obtaining simple, or at least efficient, recurrence formulas to count maps on surfaces according to their genus and size parameters. For us, a map is the 2-cell embedding of a connected multigraph in a compact connected surface, considered up to homeomorphism. Our surfaces are not necessarily orientable, and we call genus of a surface the number g ∈ 1 2 N such that its Euler characteric is 2 − 2g. The sphere has genus 0, the projective plane has genus 1 2 , the torus and Klein bottle have genus 1, etc. Perhaps one of the nicest-looking formulas in the field of map enumeration is the Goulden-Jackson recurrence formula for orientable triangulations, i.e. maps in which all faces are incident to three edge-sides. The Goulden-Jackson recurrence [GJ08], in fact also discovered in an equivalent form in [KKN99,Eq. (B.6)], asserts that the number t n,g of rooted 1 triangulations with n faces on an orientable surface of genus g is solution of the equation (n + 1)t n,g = 4n(3n − 2)(3n − 4)t n−1,g−1 + 4 i+j=n−2 h+k=g (3i + 2)(3j + 2)t i,h t j,k . (1) This formula was immediately recognized as a breakthrough in the field, because it gives a much better access to these numbers (computational or theoretical) than the classical techniques.
Indeed, in the classical approach, one introduces generating functions of maps of genus g with a certain number of additional boundaries, and one shows that a combinatorial operation of root-deletion on the maps (the "Tutte decomposition") implies a functional equation for these functions. This approach has been very successful in the planar case since the work of Tutte, see e.g. [Tut62a,Tut62b,Tut63,BC94,BBM17,BBM11]. In higher genus, it was pioneered by Lehman and Walsh [WL72] and later Bender and Canfield, who showed that the generating functions of maps of fixed genus and number of boundaries can be computed inductively on the Euler characteristic, thus revealing their particularly nice algebraic structure as well as their singular behaviour [BC86,BC91]. Bender and Canfield's inductive technique can be seen as a predecessor of the Chekhov-Eynard-Orantin topological recursion [CEO06,EO07], a powerful theory invented in the context of matrix integrals [Eyn16] which has now been applied to study the structure of fixed-genus generating functions of many models of maps or in enumerative geometry [Eyn14,ACEH20,BDBKS20,BCEGF21].
The need to introduce additional boundaries (and "catalytic" variables to mark their sizes) makes these approaches ineffective for large values of g. There seems to be no hope to obtain control on the bivariate numbers t n,g for non-fixed g in this way, a striking contrast with the recurrence (1). For example (1) also gives access to the so-called double-scaling limit of the numbers t n,g [KKN99,BGR08], and it is also crucially used in the recent Budzinski-Louf breakthrough on large genus asymptotics [BL20]. Perhaps we should insist on the fact that we mean no harm to the "classical approach". The study of the rational parametrization it gives rise to is a fascinating subject, including purely bijective combinatorics [CMS09, CD17, Lep19, AL19, DL20], with probable link to the study of random geometries [BG14,BGM21]. On the other hand, as of today, the bijective interpretation of the Goulden-Jackson recurrence (1) is wide open.
One reason why the recurrence (1) gives access to different results is because it comes from a completely different technique. It is based on the fact that the generating function of maps on orientable surfaces, with an infinite number of variables p i , i ≥ 1 (p i marking faces of degree i), is a solution of the KP hierarchy -an infinite sequence of PDEs originating from the theory of integrable systems, with deep connections to infinite dimensional Lie algebras and algebraic combinatorics [KRR13,MJD00]. The first equation of the hierarchy (the KP equation) reads −F 3,1 + F 2 2 + 1 2 F 2 1 2 + 1 12 where each i-index indicates a partial derivative with respect to p i . In order to go from the KP equation (2) to the recurrence (1), Goulden and Jackson use the fact that the generating function F (p 1 , p 2 , p 3 , 0, . . . ) of maps having only faces of sizes 1, 2, 3 can in fact be expressed in terms of the series F (0, 0, t, 0, . . . ) of triangulations only. This enables one so set p i = tδ i,3 in (2) and obtain an ODE for the generating function of triangulations. The fact that the variables p 1 and p 2 can be eliminated in this way relies on local surgery operations that can, in fact, be interpreted as first cases of the classical Tutte decomposition.
A similar elimination technique has since been used to obtain similar results for other models of maps. In [CC15], Carrell and the second author use the fact that the generating function of bipartite maps solves the KP hierarchy, and local operations related to the first Tutte equations for bipartite quadrangulations, to obtain a recurrence formula similar to (1) to count maps by vertices and faces. In [KZ15], Kazarian and Zograf use a slightly different elimination procedure, using the so-called Virasoro constraints (which are also related to Tutte decompositions) to recover the recurrence of [CC15] and to obtain an analogue for bipartite maps. These three works only use the first KP equation. Finally, Louf [Lou19] uses a different integrable hierarchy (the Toda hierarchy) to obtain a remarkable recurrence counting bipartite maps of arbitrary genus with control on all face degrees, using a different elimination technique inspired by Okounkov's work on Hurwitz numbers [Oko00].
In this paper, we are interested in obtaining variants of these results for non-oriented surfaces. Our starting point is the fact that generating functions of maps (or bipartite maps) are solutions of the BKP hierarchy of Kac and Van De Leur [KvdL98] (see also the appendix of [BCD21a]). An important difference between the KP and BKP hierarchy is that the function F which is a solution of this hierarchy also involves a so-called charge parameter N, which in our context will always be a variable marking faces or vertices of a certain kind. The first BKP equation reads; and where S 2 (N) is a model-dependant normalizing factor that will always be an explicit rational function in our case. In [Car14], Carrell used the fact that the generating function of non-oriented maps satisfies this equation, together with the elimination techniques developed by Goulden and Jackson in the orientable case, to obtain a functional equation for the case of triangulations (this technique leads to an explicit recurrence, see Theorem 4.9 below).
The first task we perform in this paper, somehow unsurprisingly, is to apply the elimination of variables from the papers [CC15,KZ15] to the BKP equation, to obtain recurrences of the same kind to count maps (by vertices and edges) and bipartite maps (by edges, and vertices of each colour) on non-oriented surfaces. The Virasoro constraints for these models are known (e.g. [BCD21a]) and our main task here is to make sure that the elimination procedure indeed works, i.e. that these equations indeed enable to reduce all derivatives appearing in (3) to differential polynomials in a single variable. For completeness, we also treat Carrell's case of triangulations explicitly. All these recurrences are larger than (1), but incredibly short compared to any alternative, and it is not unreasonable to believe that they could have a combinatorial interpretation. For example, we obtain in Section 5 the following recurrence formula. Everywhere in the paper, the symbol denotes a sum over elements of 1 2 N.
The crucial fact that the BKP equation (3) involves not only the function F (N) but also its shifts F (N + 2) and F (N − 2) has an important effect on the recurrence formulas we obtain. The functional equations corresponding to these recurrences, which involve derivatives but also shifts of variables, are not ODE in their main variable. In return, the recurrences obtained do not have polynomial coefficients (for example (4) contains binomial coefficients, which are not polynomials in the summation variables). This is a deep structural difference between the recurrences (1) and recurrences such as (4).
It is natural to ask if one could instead obtain formulas in which the shifts are not involved, i.e. true polynomial recurrence formulas, corresponding to nonlinear ODEs with polynomial coefficients for the associated generating functions. This would be much more satisfying, at least at the conceptual level. Maybe surprisingly, we will see that the answer to this question is yes. To see this, we will have to use several (in fact, three) equations of the BKP hierarchy. Using additional derivations and manipulations, we will be able to eliminate the shifts from equations, and obtain equations at fixed N, at the price of having to consider higher derivatives. It is not obvious, but it will be true, that a finite number of Virasoro constraints will still be sufficient to perform the elimination of variables in this context.
Due to the use of higher BKP equations and additional manipulations involved, the equations thus obtained are bigger than the previous ones 2 . We will only state them here in a non-explicit form. The reader eager to see them at work may access these equations, and use them to compute numbers of maps, in the accompanying Maple worksheet [BCD21b]. A typical statement we obtain from these methods is the following.
Theorem 1.2 (Counting maps by edges and genus -unshifted recurrence). The number h g n of rooted maps of genus g with n edges, orientable or not, is solution of an explicit recurrence relation of the form where the P a,b,k are rational functions with P 0,0,1 = 0, and K 1 , K 2 , K 3 < ∞.
We will obtain similar theorems for other models, in particular one for bipartite maps (Theorem 4.5), and for triangulations (Theorem 4.10). Moreover, we will in fact prove a version of Theorem 1.2 with control on the number of faces, from which we obtain a closed recurrence formula enumerating one-face maps, small enough to be explicitly written.
Theorem 1.3 (Ledoux's recursion for non-oriented one-face maps). The number u g n of rooted non-oriented maps of genus g with n edges and only one face (or equivalently with n edges 2 They are however not gigantic, see the accompanying worksheet [BCD21b]. The main ODE for maps fits in slightly more than a page in \tiny LaTex print, we have however chosen not to reproduce it here. and only one vertex) is given by the recursion (6) (n + 1)u g n = (8n − 2)u g n−1 − (4n − 1)u with the convention that u g n = 0 for g < 0 and g > n 2 and with the initial condition u The recurrence (6) was first obtained by Ledoux [Led09] using matrix integral techniques unrelated (as far as we know) to the BKP equation. It is remarkable to see that it is, in fact, the shadow of bigger nonlinear recurrence giving access to an arbitrary number of faces. The Ledoux recurrence can be viewed as an non-oriented version of the infamous Harer-Zagier recurrence, a similar (yet smaller) formula which covers the case of orientable one-face maps (and which is itself a special case of the recurrence of [CC15]). The Harer-Zagier recurrence has a nice analogue in the bipartite case due to Adrianov [Adr97], and it is natural to ask if our non-shifted recursion in the bipartite case implies an non-oriented version of Adrianov's result. The answer is yes.
Theorem 1.4 (A recurrence for non-oriented bipartite one-face maps). The number b i,j n of rooted one-face maps with n edges, i white and j black vertices, orientable or not, is given by the recursion: with the convention that b i,j n = 0 for i + j > n + 1, and b i,0 n = b 0,j n = 0 and the initial conditions b 1,1 1 = b 2,1 2 = b 1,2 2 = b 1,1 2 = 1, b 3,1 3 = b 1,3 3 = 1, b 2,2 3 = b 2,1 3 = b 1,2 3 = 3, b 1,1 3 = 4. To conclude this introduction, it is natural to ask if our techniques of shift elimination are specific to the case of maps or apply to general solutions of the BKP hierarchy. The latter is in fact true, and any function F (N) which solves the BKP hierarchy is in fact solution of an explicit (yet big) PDE involving only the function F (N) and its derivatives, with no shifts (Theorem B.1 in the appendix). We are not aware of any in-depth study of such "fixed charge" BKP equations, which might be worth considering in the future.
Structure of the paper. In Section 2, we will recall what we need about the first BKP equations, directing the reader to other sources for the depth of the BKP theory. In Section 3, we will address the case of maps, taking the time to explain the main ideas and techniques. We will write the Virasoro constraints, and show how to use them to express some derivatives of a specialization of the main BKP tau function as univariate differential polynomials. This will give us "shifted" equations. We will also show how to eliminate the shifts appearing in the BKP equation using instead the first three BKP equations to obtain non-shifted ODEs. In Section 4.1 and Section 4.2, we will address the cases of bipartite maps and triangulations. The main steps are similar to the case of maps and we will give fewer details than in the previous section. In Section 5, we apply the technique of elimination of variables of the paper [CC15] to obtain slightly different recurrence formulas than in Section 3 to count maps.
Appendix A contains tables of the numbers of rooted maps and bipartite maps of genus g with n edges and of rooted triangulations of genus g with 2n faces, generated with our recurrences. Appendix B derives the fixed charge equation for BKP solutions (Theorem B.1), which we do not use directly in this paper.
Throughout the paper, the notation R[·], R(·), R[[·]] denote respectively polynomials, rational functions, and formal power series with coefficients in the ring R.
Accompanying Maple worksheet. A Maple worksheet containing an implementation of the recurrences of this paper, together with automated calculations of the bigger ODEs for the different cases (as well as certain proofs regarding their top coefficients) is available in both Maple and html form in [BCD21b]. The worksheet also contains recursive programs obtained from these ODEs, as well tables for small genus and consistency checks against existing formulas of the literature.

A FEW WORDS ON THE BKP HIERARCHY
In this paper, we will use the BKP hierarchy as a black box, and only recall the statements and equations needed for our purposes. We refer the reader to [KvdL98,VdL01] for the general theory, and to the appendix of our previous paper [BCD21a] for details about the applications to maps and bipartite maps.
The BKP hierarchy is an infinite set of partial differential equations (PDEs) for a sequence of functions τ (N) N ∈Z depending on "time parameters" (formal variables) p 1 , p 2 , . . . . For our combinatorial purposes, it will be convenient to think of the symbol N as a formal variable rather than an integer, and this turns out to be possible under technical conditions, formalized in the notion of "formal N" BKP tau function in [BCD21a].
A formal N BKP tau-function is in fact a pair, consisting of a formal power series τ (N) ∈ Q(N)[[p 1 , p 2 , . . . ]], together with a normalizing sequence (β N ) N ∈Z which is such that for N ≥ 0, and for respectively every odd positive integer k and every positive integer k, for some rational functions R k (N), S k (N) ∈ Q(N). These conditions may seem technical but they are crucial to stating the equations of the BKP hierarchy in a formal way as we will do here. In the context of enumeration, the field Q will often be promoted to a field of rational functions or formal Laurent series involving additional variables, for example Q(t) so that The typical definition of a (formal or not) BKP tau function makes use of the infinite wedge formalism. It is the image of the orbit of the exponential of an infinite-dimensional Lie algebra, often denoted b(∞), via the boson-fermion correspondence. We refer the reader to the references mentioned above. For the purposes of this paper, we will admit the PDEs of the BKP hierarchy as a definition: Here, h j denotes the complete homogeneous symmetric function of degree j, and we define U(q) = e r≥1 qrDr andĎ = (kD k ) k≥1 , where D r is the Hirota derivative with respect to p r , By extracting coefficients in the variables q 1 , q 2 , . . . in (9), one obtains explicit PDEs for the function τ (N), which altogether form the BKP hierarchy. For example, by setting k = 2 and extracting the coefficient of q 3 , we obtain the BKP equation (3) stated in the introduction, where we recall the notation F (N) = log τ (N) and where indices indicate partial derivatives, We will only need two other equations of the hierarchy, namely the following bilinear identi- ) 2 , obtained respectively by extracting the coefficient of q 4 and q 5 , again with k = 2.
We now proceed with map enumeration.

THE CASE OF MAPS
3.1. Generating functions of maps. For us a surface is a non-oriented two-dimensional real manifold without boundary. A surface of Euler characteristic 2 − 2g has genus g. A map is a graph (with loops and multiple edges allowed) embedded in a surface such that the complement of the embedding is a disjoint collection of contractible components, called faces. The genus of the map is the one of the underlying surface. A corner of a map is a small angular sector around a vertex delimited by two consecutive edge-sides; the degree of a face/vertex is the number of corners belonging to it/adjacent to it, respectively. In this paper orientable surfaces do not play a particular role, however in some places we will explicitly use the terminology non-oriented maps to emphasize that our surfaces can be orientable or not.
We will be interested in enumeration of rooted maps, i.e. maps with a distinguished and oriented corner called the root corner. Rooted maps are considered up to homeomorphisms preserving the root corner. Define the generating function The following specialization operator plays a crucial role throughout the paper.
Definition 3.1 (Specialization θ). We let θ be the operator that specializes all variables p i to the variable z, namely θ(p i ) = z for every i ≥ 1.
We define the formal power series which is the bivariate generating function of rooted non-oriented maps M with variables t, u, z marking respectively twice the number of edges, the numbers of vertices, and faces, i.e. H i,j n denotes the number of rooted non-oriented maps with n edges, i vertices and j faces. It is important to note that a map with n edges, i vertices, and j faces, has Euler characteristic i − n + j = 2 − 2g, so the genus is implicitely controlled in this generating function and Θ(t, z, u) can be rewritten as We additionally set h g n := H g n (1, 1) for the number of rooted, non-oriented maps of genus g with n edges and u g n := H n+1−2g,1 n for the number of rooted, non-oriented maps of genus g with n edges and only one face.
The main goal of this section is to obtain functional equations on the function Θ(t, z, u), allowing us to compute its coefficients. For this, we start from the fact that the "bigger" function F has a deep structure inherited from the BKP hierarchy, which was proved by [VdL01] using a connection with matrix integrals (see also [BCD21a,Appendix] for details on the connection with maps). Here we use the notation 2p = (2p 1 , 2p 2 , 2p 3 , . . . ).
Proposition 3.2 implies that F (t, 2p, N) satisfies the BKP equation (3). It is tempting to apply the operator θ to this equation in order to get information on the function Θ(t, z, u), however because partial derivatives with respect to the p i do not commute with θ, it is not obvious that such an approach will succeed. For a sequence of non-negative integers λ = (i 1 , . . . , i k ), we introduce the quantity The F θ λ are the quantities naturally appearing when applying θ to the BKP equation (3). In order to obtain information on the F θ λ , we use the fact that τ (t, p, u) satisfies the following Virasoro constraints.
It itself comes from the homogeneity relation k≥1 p k p * k F = t ∂F ∂t by acting with ∂ n 1 +n 2 +n 3 and applying θ. This produces (16). Note that the size of all vectors indexing F θ s appearing in the RHS of (16) are strictly smaller than the size i + 2 + n 1 + 2n 2 + 3n 3 in the LHS. In order to be able to iterate this equation on all these terms, we need all vectors appearing in the RHS to have at most one part larger than 3. The only term on the RHS that could have two parts larger than 3 is of the form F θ a,b,3 n 3 ,2 n 2 ,1 n 1 . Since a + b = i, this does not happen unless i + 2 > 9. We thus obtain (17). The last statement is a direct check.
Remark 1. The Virasoro constraints in fact imply a more general result: F θ λ for any λ is a differential polynomial of Θ. This is proved by applying k j=1 ∂ ∂p i j to (18) and performing an induction on |λ|. We will refrain from writing it since we will not need it in full generality. Instead, the previous proposition is enough to cover the cases we need with explicit formulas, involving vectors of size |λ| ≤ 6.
We insist on the fact that the recurrence given in Proposition 3.4 can be fully automated to compute the polynomials P λ , and this is done in the accompanying Maple worksheet [BCD21b].
It is now immediate to see that applying the operator θ to the BKP equation (3) produces a functional equation on Θ(t, z, u).
Theorem 3.5. The generating function Θ(t, z, u) satisfies a functional equation of the following form: where P and Q are quadratic polynomials with coefficients in Q[t, u, z].
Proof. Consider (3) for τ (t, 2p, u). Applying θ to both sides, taking the derivative with respect to t and substituting the initial equation (3) back into it to eliminate exponentials, we thanks to the identity ∂ ∂t S 2 (u) = 4 t S 2 (u) implied by S 2 (u) = t 4 u(u − 1). Proposition 3.4 immediately concludes the proof.
Here we have extracted, respectively, in (19) (after substitution u → ur, z → zr), the coefficient of [t 2n+4 r n+2−2g ], of [t 2n 1 −1 r n 1 −2g 1 ], and of [t 2n 2 +5 r n 2 +2−2g 2 ], in the RHS, in the first factor of the LHS, and in the second factor of the LHS. Moreover we have used In our case m = n 1 − 2g 1 , and we parametrized k as k = n 1 + 2 − 2g 0 (the condition k ≥ m + 2 translates into g 0 ≤ g 1 , and the summand is null when g 0 < 0). It now only remains to group the terms of the form H i,j n with i + j = n + 2 − 2g. In the LHS they contribute to the first term H g n and in the RHS they appear as the terms H p,j n 1 when n 1 = n, n 2 = 0, g 1 = g, g 2 = 0, g 0 = g). Collecting these terms on the RHS gives 3 2 δ n 1 ,n δ g 1 ,g δ g 0 ,g u 2 which leads to the main equation of the theorem. The identification of the initial conditions for n ≥ 2 can be done, either: by hand drawing, or from the OEIS, or from explicit expansions in small genera using the equations of this paper, or from the expansion in Zonal polynomials up to order n = 2.

3.2.
Removing the shifts. We now proceed with the task of obtaining a functional equation on the function Θ(t, z, u) which does not involve any shift on the variable u. We will do this by using the three equations (3), (11), (12) to eliminate the shifts, and apply the operator θ. This will make terms of the form F θ λ appear, with larger partitions λ than in the previous section, but fortunately they are still in the range covered by Proposition 3.4. We have Theorem 3.7. There exists a polynomial P ∈ Q[t, u, z][x 1 , . . . , x 6 ] of degree 5 such that P ∂ ∂t Θ(t, z, u), . . . , ∂ 6 ∂t 6 Θ(t, z, u) ≡ 0. Proof. Denote ∆f (u) = f (u + 2) − f (u − 2) and ∇f = f (u + 2) + f (u − 2), and E = S 2 (u)e ∇Θ(t,z,u)−2Θ(t,z,u) .

An explicit form of P can be obtained by applying Proposition 3.4 to the following equation
Using (3), (11), (12) and applying θ we obtain the following equations , where KP1, KP2, KP3 are given in the statement of theorem. The difference between KP1, KP2, KP3 and the LHS in (3), (11), (12) is due to the fact that τ (N) is a formal N tau function of the BKP hierarchy after rescaling the variables p → 2p. Using (16) and the identity S 2 (u) = t 4 u(u − 1) we have and ∇(F θ 1 2 ) = t 6 ∂ 2 ∂t 2 + 2t 5 ∂ ∂t ∇Θ(t, z, u) + t 4 uz + t 2 u so that the third BKP equation reads We now use the first two BKP equations to express ∇ ∂ ∂t Θ(t, z, u), ∇ ∂ 2 ∂t 2 Θ(t, z, u) and ∆ ∂ ∂t Θ(t, z, u) in terms of Θ(t, z, u) and its t-derivatives. Those expressions can then substituted into (22). Taking the t-derivative of the first BKP equation, we have and another derivative gives and further From the second BKP equation, This gives (20). The statement about the form of polynomial P is a direct consequence of Proposition 3.4. Theorem 1.2 is an immediate consequence of Theorem 3.7.
Proof of Theorem 1.2. It is enough to make the change of variables u → ur, z → zr in (20) and extract the coefficient of [t 2n r n+2g−2 ]. This substitution allows to track the genus of the underlying maps. Extracting the coefficient gives a recursion for the bivariate version of h g n which additionally tracks the number of vertices and faces via u and z. Specializing u = z = 1 gives the recursion for h g n of the form (5) with a, b depending on the specific form of ODE given by (20). A direct examination of the highest degree terms of this recurrence implemented in [BCD21b] shows that h g n = h g−1/2 n − n 1 =1..n−1 n 1 +n 2 =n g 1 =0..g g 1 +g 2 =g (n 1 + 1)(n 2 + 1) 42(n + 1) h g 1 n 1 h g 2 n 2 n 1 ,...,n k ≥1 n 1 +···+n k =n−a g 1 ,...,g k ≥0 g 1 +···+g k =g−b P a,b,k (n 1 , . . . , n k )h g 1 n 1 h g 2 n 2 . . . t g k n k , which finishes the proof.
The coefficient of z 1 in Θ(t, z, u) is the generating function of maps having only one face, with control on the number of edges and vertices (equivalently, edges and genus). Extracting the bottom coefficient in z in (20), we obtain a linear ODE for this generating function. It is equivalent to Ledoux's recurrence (6) stated in the introduction.
Corollary 3.8. The generating function of rooted non-oriented maps with only one face satisfies the following linear ODE

RECURRENCES FOR BIPARTITE MAPS AND TRIANGULATIONS
4.1. Non-oriented bipartite maps. Consider the generating function where we sum over all rooted non-oriented bipartite maps, and v • (M), v • (M) denote the number of white and black vertices, respectively. Similarly, as in the case of general maps, the function G inherits a deep structure from the BKP hierarchy. This result can be derived directly from Van de Leur's work [VdL01], even though it is not stated explicitly there (see [BCD21a,Appendix] for additional details on the connection with maps).
We recall that θ(p i ) = z for i ≥ 1. Define the power series which is the generating function of rooted, non-oriented bipartite maps M. The variables t, u, v, z mark the number of edges, black vertices, white vertices and faces, respectively so that K i,j,k n denotes the number of rooted non-oriented bipartite maps with n edges, i black vertices, j white vertices and k faces (the root vertex is black by convention). Note that due to the Euler relation we can rewrite η(t, z, u, v) so that it is parametrized by the number of edges and genus: We additionally set k g n := K g n (1, 1, 1) for the number of rooted non-oriented bipartite maps of genus g with n edges and b i,j n := K i,j,1 n for the number of rooted non-oriented bipartite maps of genus g with n edges, i black and j white vertices, and only one face.
In analogy with Proposition 3.4 we express G θ λ in terms of ∂ i ∂t i η(t, z, u, v), where for a sequence of non-negative integers λ = (i 1 , . . . , i k ).
An explicit form of Q can be obtained by applying Proposition 4.2 to the following equation The proof is analogous to the proof of Theorem 3.7 and left to the reader. We have two immediate corollaries, Theorem 4.6 which is analogous to Theorem 1.2 for bipartite maps, and Theorem 4.7 which is a bipartite analogue of Ledoux's recurrence and a non-oriented analogue of Adrianov's.
Theorem 4.7 (A recurrence for non-oriented bipartite one-face maps). The number b i,j n of rooted one-face maps with n edges, i white and j black vertices, orientable or not, is given by the recursion: Extracting the coefficient of [t n u i v j ] produces the desired recursion.

4.2.
Non-oriented triangulations. The generating series of triangulations can be obtained from F (t, p, u) (given by (13)) by applying another specialization instead of θ. Indeed, define the specialization operator θ 3 by θ 3 (p i ) := zδ 3,i . This operator enforces that all faces must be of degree 3, and is the generating function of rooted, non-oriented triangulations M. Of course, triangulations satisfy 2e(M) = 3f (M). By using Euler's relation, one can expand Ξ(t, z, u) by the genus and the number of edges, and here t g n denotes the number of rooted, non-oriented triangulations with 3n edges (or equivalently 2n faces) and genus g.
Similarly as in the previous sections, we want to express F θ 3 λ as a polynomial in Ξ(t, z, u) and its derivatives with respect to t, where for a sequence of non-negative integers λ = (i 1 , . . . , i k ).
Since the sizes of the vectors indexing F θ 3 s appearing in the RHS of (32) are strictly smaller than |λ|, one computes F θ λ recursively for vectors of the form λ = [ℓ, 3 n 3 , 2 n 2 , 1 n 1 ], where ℓ ≤ 10 (by eliminating all the parts of length 3 thanks to (33), and reducing the sizes of the indexing vectors thanks to (32) and finally using the recurrence (34) for the parts of the form F θ 1 l ). The last statement follows by induction on ℓ.
Remark 2. We want to highlight the fact that the above computations are possible because we are working with the specific model of triangulations. Replacing the specialization θ 3 by θ l : p i → δ i,l with l ≥ 4 makes the above technique fail to even compute F θ l 1 .
Proof. As in the case of bipartite maps, the proof is almost identical to the proof of Theorem 3.6 and we leave the details for the interested reader. The only difference is that (19) should be replaced by Theorem 4.10. There exists a polynomial R ∈ Q[t, u, z][x 1 , . . . , x 6 ] of degree 5 such that R ∂ ∂t Ξ(t, z, u), . . . , ∂ 6 ∂t 6 Ξ(t, z, u) ≡ 0.
The proof is the same as in the previous cases, so we leave it as an exercise. As a standard consequence we have.
In analogy to what we did for maps and bipartite maps, it would be natural to study now the case of triangulations with only one vertex (or by duality, cubic one-face maps). However, there exist very explicit and simple formulas in this case, obtained from bijective methods [BC11] so we prefer not to go into such calculations here.

ANOTHER METHOD IN THE CASE OF MAPS
In this section, we quickly adress the case of maps treated in Section 3 with another method, which actually leads to different recurrence relations. The situation is similar to the orientable case, where the approaches used in [CC15] and [KZ15] differ. In Section 3 (non-oriented analogue of [KZ15]) we started from the fact that the generating function F of maps is a BKP tau-function, and applied the substitution operator θ : p i → z. In this section, we will instead start from the fact that the generating function G of bipartite maps is a BKP tau function, and apply the different substitution operator θ 2 : p i → δ i,2 . We will only treat the equations with shifts, our main motivation being that they are relatively nice looking -Proof. The lemma can easily be proved with Virasoro constraints in the same manner as Proposition 4.8 and details are left to the reader. However, a calculation-free proof based on digon contraction and elementary combinatorial map operation is also easily doable. The proof is completely similar to [CC15, Lemma 7], the only difference is the extra term of genus g − 1/2 in the two equations involving a hexagonal default (i.e. a p 3 -derivative). This term comes from the possibility to create a rooted quadrangulation by adding a twisted diagonal inside a digon. This is the only difference between the oriented and non-oriented case, and it adds one term to Equation (11) in [CC15]. Once this difference is taken into account, the proof of [CC15, Lemma 7] can be copied verbatim.
A direct consequence of what precedes is the recurrence formula stated as Theorem 1.1 in the introduction. One can also obtain a version with control on vertices and faces, from which Theorem 1.1 follows immediately.
We finally equate the RHS of the above two equations and multiply by KP1 3 to obtain (42). Email address: mdolega@impan.pl