Enumeration of unrooted odd-valent regular planar maps

Transcription

1 Enumeration of unroote o-valent regular planar maps Zhicheng Gao School of Mathematics an Statistics Carleton University Ottawa, Ontario Canaa K1S 5B6 Valery A. Lisovets Institute of Mathematics National Acaemy of Sciences Mins, Belarus Nicholas Wormal Department of Combinatorics an Optimization University of Waterloo Waterloo ON Canaa N2L 3G1 Abstract We erive close formulae for the numbers of roote maps with a fixe number of vertices of the same o egree except for the root vertex an one other exceptional vertex of egree 1. The same applies to the generating functions for these numbers. Similar results, but without the vertex of egree 1, were obtaine by the first author an Rahman. We also show, by manipulating a recursion of Bouttier, Di Francesco an Guitter, that there are close formulae when the exceptional vertex has arbitrary egree. We combine these formulae with results of the secon author to count unroote regular maps of o egree. In this way we obtain, for each even n, a close Supporte by University of Macau an NSERC. Supporte by the Belarusian RFFR (grant No. F Supporte by the Canaa Research Chairs program, NSERC an the University of Macau. 1

2 formula for the function f n whose value at o positive integers r is the number of unroote maps (up to orientation-preserving homeomorphisms with n vertices an egree r. The formula for f n becomes more cumbersome as n increases, but for n > 2 each has a boune number of terms inepenent of r Mathematics Subject Classification. Primary: 05C30, Seconary: 05A15. Key wors. o egree, unroote map, roote planar map, regular map, rotation, quotient map, close formula 1 Introuction 1.1 Motivation In the late 1970 s the secon-name author evelope a metho of counting unroote planar maps [Lis81] (see also [Lis85, Lis98]. It relies on the concept of the quotient map of a (geometrically implemente map with respect to a rotational automorphism. The iea is to reuce the problem to the enumeration of roote maps of the same type an the arising quotient maps. The metho turne out to be effective for iverse classes of maps (see [Lis04]. In particular it is quite applicable for counting unroote vertex egree specifie maps, although the resulting reuctive formulae are necessarily fairly cumbersome. But in general we encounter a more serious obstacle in applying this approach: the enumeration of the arising roote egree specifie maps. If the maps uner consieration are eulerian, that is even-valent, then all the quotient maps are eulerian or unicursal (the latter term means that only two vertices are of o egree, an there are remarable sum-free formulae iscovere by Tutte [Tut62] for counting roote eulerian or unicursal maps with a given vertex egree istribution. In particular, this enable the secon author to count unroote regular maps of even egree [Lis85]. Sum-free close formulae are appealing goals for research in enumeration. However, in the general case for maps with o-egree vertices such simple formulae are not nown (see [BenC94], where a certain general but inconvenient formula is given. So the corresponing reuctive formulae are impractical for maps having many o-egree vertices. At present, the only nown notable exception is a simple close formula for the number of roote 3-regular maps [Mul66]. Accoringly, there is a simple formula for the number of unroote 3-regular maps. This was publishe (with a minor error in [LisW87]. A technique evelope by the first author [Gao93] (cf. [GaoR97] shows that there is another class of effectively enumerable roote maps with many o-egree vertices, namely, r-regular maps with a fixe number of vertices. One special case of this has alreay been treate: the case of two vertices, by Bousquet, Labelle an Leroux [BouLL02]. However, in orer to be applicable for counting unroote maps of the same classes, the roote enumeration shoul be extene to the maps of a similar specification but possessing one or two aitional o-egree vertices. One of the main aims of the present paper is to o this for the o-valent regular maps (i.e. maps with all vertices of o egree. We use two methos for this. The first achieves simple close formulae, if we permit one extra vertex of egree 1 an another of arbitrary egree. Even this requires consierable effort an a nontrivial extension of the technique in [GaoR97] to 2

3 be able to eal with truncations of power series which somewhat mysteriously cancel with other nice functions (see (3.28. We obtain simple formulae for the generating functions in this way. Further extensions along these lines loo ifficult. The other main aim is to translate these results to the enumeration of unroote regular maps with n vertices all of o egree r. The results on roote maps mentione above o not cover the cases when n > 2 an gc(r, n 2 has a factor other than 1 an r. To cover these cases, we use a secon metho for the roote maps. We show that recursions erive by Bouttier, Di Francesco an Guitter [BoutDG02] can be manipulate an expane in such a way to obtain close formulae for the misse cases. These cases involve a fixe number of vertices of arbitrary egree r, together with up to two other vertices of egree that is a fixe fraction of r. However, this metho oes not reveal a simple formula for the generating function, as obtaine using the first metho. In fact, the expansion technique we use can hanle any fixe number of vertices of arbitrary egrees in this manner. As a result of all this, we show that there exist close formulae for the numbers of unroote regular maps with a given number n of vertices as a function of the o egree r. In Section 6 we give computational results: formulae for the functions of r (for small n, an some actual numbers for small n an r. 1.2 Main efinitions an notation A planar map is a 2-cell embeing of a planar connecte graph (loops an multiple eges allowe in an oriente sphere. We consier only planar maps, therefore for brevity we will merely call them maps hereafter. A map is roote if one of its ege-ens (nown also as ege-vertex incience pairs, arts, semi-eges, or brins in French is istinguishe as the root. The corresponing vertex is calle the root-vertex. It is well nown an important that no non-trivial automorphism of a map leaves the root fixe. The egree of a vertex is the number of ege-ens incient with it. This is often calle valency in the map enumeration literature. We retain this terminology in the following form: an o-valent map is one with all vertices of o egree. In this paper, counting unroote maps means counting up to orientation-preserving homeomorphisms. There is also a metho of counting up to all homeomorphisms given by the thir author [Wor81], but this is in general very recursive in nature an not so convenient for obtaining close formulae. If K(m is a class of maps with m eges, then K + (m enotes the number of unroote maps in K(m an K(m enotes the number of roote maps. Hence K + (m = K(m = Γ K 1, an it is clear that K(m = Γ K 2m Aut(Γ, where Aut(Γ is the orer of the automorphism group of the map Γ. We efine A(1 a 1 2 a 2 3 a3 ; m to be the class of planar maps of the inicate vertex egree specification, that is, with a i vertices of egree i; the implie number of eges, m = 3

4 (a 1 + 2a 2 + 3a 3 + /2, is also recore in the notation. These maps have vertices an (by Euler s formula v = a i f = m v + 2 (1.1 faces. We call maps of such a specification r-regular if a i = 0 for i r. Moreover, we nee to consier maps in which one or two vertices of egree 1 are istinguishe as singular; a map cannot be roote at a singular vertex. The presence of singular vertices is inicate by the corresponing number of asteriss ( in the subscript. Singular vertices play the role of axial elements in a rotational automorphism of a covering of the roote map (see below. 2 Reuctive enumeration of unroote regular maps In this section we relate the number A + (r n = A + (r n ; m of unroote r-regular maps with n vertices an m = nr/2 eges to the numbers A(r n ; m of the corresponing roote maps, an roote maps of certain relate types. The technique evelope in [Lis81, Lis85] will be use; these references shoul be consulte for further etails of the general iscussion in this section. 2.1 Quotient maps an liftings Consier a map Γ an its rotational automorphism α of orer ρ > 1. Then α is etermine by two axial cells (poles an the rotation angle 2πl/ρ, where l, 1 l < ρ, is prime to ρ. Now there is a unique quotient map = Γ/α. It is efine topologically an can be constructe geometrically by cutting the unerlying sphere into ρ ientical sectors, taing one of them an glueing it into a new sphere. Conversely, given ρ an two arbitrary axial cells, neither of which may be an ege, is lifte into a unique map Γ. In orer to use this approach for the enumeration of unroote maps Γ K(m, one nees first to properly ifferentiate all possible rotation axes by types to ensure a certain uniformity of the quotient maps an of their liftings. Namely, a classification W = {ω} of the axes is proper if the number of ways to lift a quotient map bac into K(m (in other wors, this is the multiplicity of the appearances of as a quotient map epens only on ρ an the type ω of the axis with respect to which has been obtaine. We enote this number by ψ ω (ρ. Further the corresponing classes of quotient maps shoul be escribe thoroughly an enumerate as the roote ones: K + (m is easily expresse in terms of these numbers multiplie by ψ ω (ρ an summe over all possible ρ an ω. Typically the classification of axes into types accoring to the nature of the two axial cells (face face, face vertex, etc. suffices. This is inee true for the present application. Constraints on ρ, together with the numbers of the corresponing quotient maps, are escribe for each type in Table 1 (where, an thereafter, for brevity, (, enotes the greatest common ivisor. Recall that m = nr/2, an note that n is always even since r is o. 4

5 Table 1: Quotient maps for A(r n ; m Type ω Constraint := P ω (ρ #(roote quotient maps := A[ω](ρ face face ρ n/2 A ( r n/ρ ; m ρ face vertex ρ (r, n 1 A ( ( r ρ 1 r (n 1/ρ ; m ρ ( face ege ρ=2 (n/2 + 1 A 1 1 r n/2 ; m+1 2 vertex vertex ρ (r, n 2 A ( ( r ρ 2 r (n 2/ρ ; m ρ ege ege ρ=2 n/2 A ( 1 2 r n/2 ; m In orer to explain these ata we note that the egrees of the axial cells are iminishe ρ times in the quotient maps, while the non-axial cells preserve their egrees. On the other han, ρ non-axial cells turn into one cell, whereas every axial cell of the original map turns into one axial cell of the quotient map. If the axial cell of Γ is an ege (in which case ρ = 2, it turns into an ege ening in a special aitional vertex of egree 1. This vertex of the quotient map is calle singular. A singular vertex is necessarily axial, an the quotient map cannot be roote at a singular vertex. Finally, the quotient map contains m/ρ eges if it has no singular vertices, an apart from this, every singular vertex contributes an aitional term of 1/2 to the number of eges of the quotient. The constraints in the secon column of Table 1 arise from the conition that all parameters of the quotient maps are integers. When ω is face face, we have ρ (n, nr/2 an so ρ n/2 since r is o. When ω is face ege, ρ = 2 (n, nr/2 + 1/2 an so n/2 must be o. The other cases shoul be obvious. Note that ω cannot be of the form vertex ege because ρ woul then be 2 (because of the axial ege but woul also have to ivie the egree of the axial vertex, which is o. So this sixth type ω oes not occur for these maps. Table 2 contains the values of number ψ ω (ρ of amissible choices of the axial cells in the quotient maps. The type is efine as in the lifte map. Here f is the number of faces, a function of other parameters as etermine by (1.1. The number of faces of the quotient maps (the secon column is significant for calculating ψ only if at least one axial cell is a face. 2.2 Reuction Now we can establish the esire uniform reuctive formula for the number of unroote regular maps. Theorem 1 A + (r n = 1 [ A(r n ; m + 2m 2 ρ mφ(ρ ω ] δ Pω(ρ ψ ω (ρ A[ω](ρ, (2.1 5

6 Table 2: Choice of axial cells in quotient maps for lifting Type ω #(faces #(axes choices := ψ ω (ρ face face face vertex face ege f 2 ρ + 2 f 1 ρ + 1 f+1 2 vertex vertex 1 ege ege 1 ( f 2 ρ f 1 ρ + 1 f ( f 2 ρ + 1 /2 where m = rn/2, φ(ρ = {i : 1 i ρ, gc(i, ρ = 1} is the Euler totient function, the secon sum is taen over the five axis types represente in the first column of Table 1, P ω (ρ an A[ω](ρ are the corresponing constraint an roote enumerator represente, respectively, in the secon an thir columns of the same table, the values of ψ ω (ρ are taen from the last column of Table 2, where the parameter f is etermine by formula (1.1 an δ Pω(ρ is the characteristic function: δ P = 1 if the conition P hols an δ P = 0 if P is false. Proof: Immeiately from the foregoing tables an formulae accoring to the general enumerative scheme for unroote planar maps [Lis81, Lis85]. Since the term A(r n ; m is the number of r-regular roote maps for which a formula was erive in [GaoR97], it remains to obtain close enumerative formulae for the functions A[ω](ρ appearing in the right-han sie of (2.1. This will be one in the rest of this paper. We can state the form of the answer as follows. Theorem 2 Let n > 2 be fixe an let enote (r 1/2. Then A + (r n is expressible as a polynomial in ( ( 2 an the quantities 2 /i δi r /i (for all o i > 2 iviing n 1 or n 2 with coefficients being polynomials in. Some cases of this theorem are treate in Sections 3 an 4 (for rational functions of rather than polynomials; see Section 3.2. The theorem is prove in full in Section 5.2. For n = 2, formulae (83 (the o half an (38 of [BouLL02] yiel A + (r 2 = 1 2 ( ( ρ 2+1 ( 2 2 /ρ φ(ρ. /ρ Along the way, we will reerive this formula. A list of explicit formulae for A + (r n for all even n 10 is given in Section 6. 3 Near-regular maps with a egree 1 vertex This section eals with roote maps which have all but the root vertex of egree r, an one other vertex of egree 1. These are important for evaluating formula (2.1. 6

7 We immeiately switch to the uals of the maps to be more in line with the arguments in [GaoR97]. Let M {r},m enote the number of roote maps with all internal faces of egree r, with faces altogether (incluing the unboune face, an m eges. Similarly, M {r}[h],m enotes the number of roote maps with all internal faces of egree r except for one istinguishe face of egree h, with faces altogether (incluing the unboune face, an m eges. From all these notations, when r is unerstoo it can be omitte; thus M [h],m = M {r}[h],m ; but if r has ifferent values in two parts of the same formula it will be inclue. We efine the generating functions M (x = m 0 M,m x m, M [h] (x = M [h],m xm. m Reuction to M,m an M [1],m We nee to express the entries A[ω](r in Table 1 in terms of M,j an M [1],j. In the following we use = 2r + 1 as before. For all but the fourth type (vertex vertex, we have (see justification below A(r n/ρ ; m/ρ = M n/ρ,m/ρ = M,+/2 ( = n/ρ (3.1 A((r/ρ 1 r (n 1/ρ ; m/ρ = nm 1+(n 1/ρ,m/ρ = nm, +(+r/ρ 1/2 ( = 1 + (n 1/ρ (3.2 A (1 1 r n/2 ; (m + 1/2 = M [1] 1+n/2,(m+1/2 = M [1], +/2 ( = 1 + n/2, (3.3 A (1 2 r n/2 ; m/2 + 1 = (rn/4m [1] 2+n/2,(m+2/2 For the vertex vertex type, if ρ = r n 2 an n > 2 an if n = 2 = (rn/4m [1], 2+/2 ( = 2 + n/2. (3.4 A((r/ρ 2 r (n 2/ρ ; m/ρ = (n/2m [1] 2+(n 2/r,n/2 = (n/2m [1], 2+/2 ( = 2 + (n 2/r (3.5 A((r/ρ 2 r (n 2/ρ ; m/ρ = M {r/ρ} 2,m/ρ, (3.6 whilst in any other case for the vertex vertex type (where n > 2 an ρ < r there is no reuction to these generating functions. Thus, nowing M an M [1] is not sufficient to obtain a final result if n > 2 an gc(r, n 2 has any factors (which are potential values of ρ other than 1 an r. The numbers n for which this can happen are those for which n 2 is not a power of 2. Since r is o, the require conition for this metho to wor can be restate as r is an o prime or gc(r, n 2 = 1 or n = 2. (3.7 To justify equations (3.1 (3.6, we consier the uals of the maps counte by the M s. Then faces of egree r become vertices of egree r. The equation for the first type 7

8 is immeiate. For the secon, the quotient map has nr/ρ rootings, of which r/ρ are on the singular vertex, maing a n-to-1 corresponence between the maps require an the ones counte by M 1+(n 1/ρ,n/ρ (where the vertex of egree 1 must be the root vertex. For the thir type, we coul have use M with a similar ajustment, but by using M [1] no ajustment is require. For the fourth type, M [1] counts maps with the root at a vertex of egree 1, so the ajustment factor is the number of possible rootings of the quotient map, ivie by 2 (as there are two vertices of egree 1. As the quotient map has nr/2ρ = n/2 eges, the factor is n/2. For the fifth case, the argument is similar but the special vertices isallow two possible rootings in the maps counte by A. 3.2 The form of M,m an M [1],m Close formulae for the numbers M,m were compute in [GaoR97], so for evaluating (3.1 (3.6 the only new results require are for M [1],m. We first escribe what in of functions will appear in the argument. In particular, the generating function for roote plane trees is well nown: M 1 (x = M {r} 1 (x = 1 1 4x. (3.8 2x We will wor with the set S[X] of polynomials in X an X 1 whose coefficients are rational functions of. It follows easily from the equations in [GaoR97] that where For M [1] (x we will prove a similar result. Lemma 1 For 2, M (xx 1/2 x ( 1 ( 2 1 S[X] (3.9 X = (1 4x 1. (3.10 M [1] (xx 1/2 x ( 2 ( 2 2 S[X]. This lemma is prove at the en of this section, along with the specific calculation of M [1] (x. The following result is obtaine irectly from (3.9 an Lemma 1 by consiering the extraction of the appropriate coefficient of x. Note that the coefficient of x i in any half-integer power of X is a polynomial in ( 2 with coefficients being rational functions of. Details of this in of calculation appear in Section 3.7. Corollary 1 For each fixe, M,+/2, M [1] [1], +/2 an M, 2+/2 are all polynomials in ( 2 with coefficients being rational functions of, an M, +t is a polynomial in ( ( 2 an 2t t with coefficients being rational functions of an t. Note that most cases of Theorem 2 (for rational functions rather than polynomials follow immeiately from Corollary 1, (3.1 (3.6 an Theorem 1. 8

9 3.3 A recursion for M [h] (x Our point of access is a recursive type relation analogous to [GaoR97, Lemma 1]. Notation For i 0 an a generating function A(x = m 0 c mx m, let R i (A(x enote the part of A(x consisting of terms with exponents at least i, that is R i (A(x = m i c m x m. Lemma 2 For 3 we have M [h] (x = x (2 1 4x 1 j=2 M j (xm [h] +1 j (x + R a/2(m [h] 1 (x + R (a+δ/2(m 1 (x where a = ( 2r + h 1 an δ = 1 if h = 1 an 0 otherwise. Proof: The common egree r is constant in the following argument. Using the type of ecomposition in [GaoR97], let M be a map of the type counte by M [h] (x, with m eges. There are two cases if the root ege of M is elete. (As usual for such arguments, the new map(s resulte are roote in canonical ways. In the first case, two maps result. If the one containing the istinguishe face has j faces then the other has + 1 j faces. The generating function for these is 2xM j (xm [h] +1 j (x. In the secon case, eleting the root ege merges the ajacent face with the root face, an there are two subcases. In the first subcase, the ajacent face has egree r, in which case the number of possibilities for the map prouce is M [h] 1,m 1, an in the secon, the ajacent face is the istinguishe one of egree h, so the new map is counte by M 1,m 1. However, in the first of these cases the egree of the root face of M must be at least 1, implying (summing the face egrees that 2m ( 2r + h + 1, an so m 1 (( 2r + h 1/2. Conversely, for m 1 in this range, the reverse operation can be carrie out in a unique way. On the other han, in the secon case, we obtain the same constraint if h 2, but the stronger constraint m 1 (( 2r + h/2 in the case h = 1 since the egree of M must be at least 2 when the root ege encompasses a face of egree 1. (For = 2, if it ha been permitte, the secon case woul have the same restriction as the first one. Thus M [h] (x = j=1 2xM j (xm [h] +1 j (x + x an the lemma follows upon solving for M [h] factor. m (( 2r+h 1/2 (M [h] 1,m xm + M 1,m x m, an using (3.8 to simplify the leaing In further wor we consier only the case h = 1. Since the number of eges in maps counte by M 1 (x is at least ( ( 2( /2 = (a + 1/2, the following is immeiate. 9

10 Corollary 2 M [1] (x = x (2 1 4x 1 j=2 M j (xm [1] +1 j (x + R a/2(m [1] 1 (x + M 1(x Now for some computations for small values of. As a founation in aition to (3.8 we use the following formula of Tutte [Tut62] (see also equation (1 in [GaoR97] for just two faces, the interior one of egree r (even or o: M {r} 2,m = m r/2 2m ( 2m 2 r/2 m r/2 ( 2 r/2 r/2. (3.11 We also mae use, in one of our alternative erivations of M [1] 3, of Tutte s formula in the case that the non-root face egree is even: M {2},m (m 1!(m ( 1 = ( 1!(m + 2! ( 2m 2( 1 m ( 1 ( (3.12 For later use, note that (3.11 implies for the generating function (see [GaoR97, Equation 13] ( 2 ( m 2m 2 M 2 (x = x m (3.13 2m m m +1 an hence From [GaoR97] we also have M 2(x = M 3 (x = ( 2 x. (3.14 (1 4x 3/2 ( 2 2 x 2+2 (1 4x 3/2. (3.15 We henceforth assume that r is an o integer, an write = (r 1/2 so that r = 2+1. The value of a/2 in Lemma 2 is then, in the case h = 1, given by a/2 = ( 2 + /2 1. (3.16 Formulae such as (3.14 an (3.15 we will call close since they are simple explicit formulae for the generating functions. In the present section, more specifically, a close expression means that it is a sum of terms involving the power of x an (1 4x where the number of terms epens only on the number of vertices, not on. In general, our aim woul be to fin such close expressions for M [h] (x in general, though we only succee in the case h = 1. For the next two calculations, we explore a irect combinatorial argument rather than the recursion in Lemma 2. (However, note that we will later evelop a general recursion that also covers the case of M [1]

11 3.4 M [1] 2 Start with a roote plane tree with m 1 eges an insert a loop into any of its 2m 2 corners, or insert the loop into the root corner an transfer the rooting onto the loop. This shows that M [1] 2,m = (2m 1M 1,m 1 as M 1,m is the number of roote planar trees with m eges. Thus, using (3.8, we have regarless of r. 3.5 M [1] 3 M [1] 2 (x = x 1 2, (3.17 A map Γ of this sort can be obtaine by inserting a loop into another map, Γ. For the loop to be inserte into the unboune face, there are 2m 2 r corners available, an we also permit the possibility of shifting the rooting onto the loop if the root corner is use. In this case Γ is counte by M 2,m 1 = M 2,m 1. {r} For the loop to be inserte in the interior face, there are r 1 corners as Γ is counte by M 2,m 1. {r 1} Hence M {r}[1] 3,m = (2m r 1M {r} 2,m 1 + (r 1M {r 1} 2,m 1. Recalling that r = we may apply (3.11 an (3.12 to obtain ( ( ( M [1] 2 2m 2 2 (m 1 2 3,m = + m 1 m 1 ( ( 2 2m 2 2 = (m 1 m 1 Note that, comparing with (3.11, we have (m 1 m 1. (3.18 M [1] 3 (x = 2x 2 M 2(x, (3.19 a relation which it woul be nice to erive via a irect bijection between the maps counte by M [1] 3 (x an those counte by M 2(x (with one less ege an with a istinguishe corner. Unfortunately, simply saying that the loop is inserte into the istinguishe corner oes not apply, since the loop cannot be inserte into a corner in the interior face without estroying the egree conition, an also the M 3,n maps with a root on the loop must be prouce somehow. 3.6 M [1] 4 The general metho in the next Section is quite complicate so we introuce it by ealing with the case of M [1] 4 separately using a very relate approach. This permits the reaer to see the central iea in a simpler context. From Corollary 2 an using (3.16, ( 3 M [1] x 4 (x = 2M j (xm [1] 1 4x 5 j (x + R 2+1(M [1] 3 (x + M 3 (x. (3.20 j=2 11

12 Of the terms in the summation, by (3.19. Also by (3.15 an (3.17 ( 2M 3 (xm [1] 2 2 (x = 2 For the next term, note that, using (3.19, 2M 2 (xm [1] 3 (x = 2x 2 ((M 2 (x 2 ( x 2+2 (1 4x 3/2 ( R 2+1 (M [1] 3 (x = R 2+1 (2x 2 M 2(x = 2x 2 R 2 1 (M 2(x Aing (3.21 an (3.23 using (3.11 gives x 1. 2 = 2x 2 (R 2 (M 2 (x (3.22 = 2x 2 (R 2+1 (M 2 (x + x 2 M 2,2. (3.23 2x 2 ((M 2 (x 2 + R 2+1 (M 2 (x + 4x 2+1 ( 2 It is shown in [GaoR97] just after (18 that (M 2 (x 2 + R 2+1 (M 2 (x = an so putting this all together we have from (3.15 an (3.20 which is ( ( (2 x 2x 2 1 4x ( 2 2 x 2+1 (1 4x 1, ( ( 2 2 x 2+1 (1 4x 1 + 4x ( 2 ( 2 +2 x 2+2 (1 4x 3/ x x 1 4x ( 2 ( 2 2 x 2+2 (1 4x 3/2 2 ( 2x 2 ( (2 + 1x 2 (1 4x 1 + 4x 2+1 (1 4x 2 + 4x i.e. ( +2x 2+2 (1 4x 3/ x 1 + x 2+2 (1 4x 3/2 2 x x ( 2 2 ( 2x ( (2 + 1(1 4x 1 + 4x(1 4x 2 + ( +2x(1 4x 3/ x 1 + x(1 4x 3/2 2 12

13 i.e. i.e. x x ( 2 2 ( 2x ( (2 + 1(1 4x 1 + 4x(1 4x 2 + +x(1 4x 2 M [1] 4 (x = x x ( 2 2 ( + 2x(2 + 1(1 4x 1 + x(8x + 1(1 4x 2. ( Higher M [1] In the formula for M [1] 5 from Lemma 2 we have close expressions plus 2M 2 (xm [1] 4 (x + 2M 3 (xm [1] 3 (x + R 3+3/2 (M [1] 4 (x. (3.26 The mile term is close. Now in general 2M 2 f(x + R (f(x (3.27 is close when f(x = (1 4x α for α a half-integer. To be precise, we will calculate H j (x for integers j 1 such that ( j 3/2 F((1 4x j 1/2 = ( 4x H j (x ( where F is the linear operator efine as First, ifferentiating (3.24 we have F(f(x = 2M 2 (xf(x + R f(x. (3.29 2M 2 (xm 2(x + R 2 (M 2(x = = ( 2 2 (x 2+1 (1 4x 1 ( 2 2 ( (2 + 1x 2 (1 4x 1 + 4x 2+1 (1 4x 2. So using (3.14 an iviing by ( 2 x, an then using ( 5/2 1 ( 4 = 2(2+1 3 etermines H 1 in (3.28: H 1 (x = ( 2, this 3( x 2(2 + 1(1 4x. ( Next, note that ( j 3/2 R x(1 4x j 3/2 = xr 1 (1 4x j 3/2 = xr (1 4x j 3/2 +x ( 4x

14 an hence F(x(1 4x j 3/2 = x2m 2 (x(1 4x j 3/2 + R x(1 4x j 3/2 ( j 3/2 = xf((1 4x j 3/2 + x ( 4x 1. 1 So, using linearity of F, for all integers j, ( j 3/2 ( 4x H j (x = F((1 4x j 1/2 1 Writing = F((1 4x(1 4x j 3/2 = F((1 4x j 3/2 + 4F(x(1 4x j 3/2 ( j 3/2 = (1 4xF((1 4x j 3/2 + 4x ( 4x 1 1 [ ( ( ] j 5/2 j 3/2 = (1 4x H j+1 (x ( 4x. 1 1 B(x, y = we may rewrite the equation above as ( x 1/2 + y 1 B( j 2, 1 H j (x = X B( j 1, 1 H j+1(x 1 with X as in (3.10. Now it is easy to chec that for all integers p an q an thus B(p 1, q B(p, q = 2p 1 2 2q 2p 1 H j+1 (x = X 2j + 3 2j (H j(x + 1. Starting with (3.30, which may be rewritten as H 1 = X(X + 2 2(2 + 1, (3.31 (3.32 we may use this recurrence to compute H j recursively for all fixe integers j, both positive an negative. Moreover, we may write H j (x = H j (X (3.33 where H j is a polynomial in X an 1/X whose coefficients are rational functions of with a boune number of terms for fixe j. The resulting recurrence for H j can be solve using stanar techniques to give H j = (3/2 1 (j + 3/2 1 X j H0 + j 1 1 (i + 3/2 1 X j i, (3.34 (j + 3/ i=0

15 where H 0 = X/(2, an (a b = a(a + 1 (a + b 1 is the rising factorial. We may chec that (3.34 is also vali for negative integers j, in which case the summation symbol is interprete as 1 i=j. We also nee to truncate at some other places. For b 0, efine F b (f(x = 2M 2 (xf(x + R b f(x (3.35 (so in particular, F = F. To simplify things a little, let us efine { M [1] (x = M [1] (x for > 2 M [1] 2 (x + 1 = for = x (3.36 Then the lone term M 1 (x in Corollary 2 is amalgamate into the term in the summation for j = 1, an we have for 3 M [1] (x = x 1 (2 M j (x 1 4x = j=2 x 1 (2 M j (x 1 4x j=3 by (3.16. From (3.34 we have for fixe j that [1] M +1 j (x + R [1] a/2( M 1 (x [1] M +1 j (x + F [1] ( 2+/2 1( M 1 (x (3.37 H j (x S[X], where S is efine as in Section 3.2. It is easy to chec that ( 2 B(0, 0 = ( 4, (3.38 an thus from (3.32, an a similar formula for B(p, q 1/B(p, q, for any fixe integers p an q ( 2 ( 4 B(p, q/ is a rational function of (3.39 an hence by (3.28 for each fixe j. F((1 4x j 1/2 ( 2 x S[X] (3.40 Proof of Lemma 1 We claim that (3.37 implies by inuction that for 2 L (X := [1] M (xx 1/2 x ( 2 ( 2 2 S[X], (3.41 which immeiately implies the lemma. 15

16 This is true for = 2 by (3.17. Assume L 1 (X ( 2 3S[X]. Then F ( 2+/2 1 ( M [1] 1 (x = x( 3 F +/2 1 (L 1 (XX 1/2. This can be evaluate by noticing that F +/2 1 is linear an also F +/2 1 (X j+1/2 = F (X j+1/2 + /2 3/2 i= x i [x i ]X j+1/2 where square bracets enote the extraction of a coefficient. Thus by (3.28 F ( 2+/2 1 ( M [1] 1 (x = x( 3 j Q(j, [X j ]L 1 (X (3.42 where Q(j, = ( j 3/2 1 ( 4x H j (x + = ( 4x B( j 1, 1 H j (X + x ( 2 S[X] + /2 3/2 i= /2 3/2 i=0 x i [x i ]X j+1/2 ( 4x +i B( j, i (3.43 by (3.39 an inverting (3.10 to x = (1 1/X/4. We now obtain (3.41 recursively from (3.37, (3.9 an (3.42. Computational note The coefficients of L (X can be obtaine from the equations referre to in the last line of the above proof together with (3.38, (3.32, (3.34 an (3.33. This gives the value of M [1] for 3 by the efinition of L in ( Computations using M an M [1] Computations in this section were basically all one using Maple. We are aiming for evaluation of the right han sies of equations ( , an for this we fin the generating functions M (x an M [1] (x for small. Note that the coefficients of M 2(x are given in (3.11 an we o not have a close formula for the generating function. For 3, we give the formulae for M (x compute using the equations in [GaoR97], which clearly emonstrate the general form of these functions. The first three cases are as follows; we compute these up to = 20 in orer to obtain our later results for maps with up to 20 vertices. M 3 (x = 1 ( 2 2 (X 1 2 x 2 16 X M 4 (x = 1 ( 3 2 (X 1 2 (2 + X 1 x 3 64 X 16

17 M 5 (x = ( 4 2 (X 1 3 ( X 18 X X 2 x 4. X 3/2 Similarly, M [1] (x can be compute for small as escribe at the en of Section 3. The functions L (X efine in (3.41 are easily erive from these. Recall that M an M are the same except when = 2. For up to 5 we have the following; more cases were compute for obtaining the numbers given later. M [1] 5 (x = M [1] 3 (x = 1 8 M [1] 4 (x = 1 ( 2 64 ( 2 M [1] 2 (x = 1 2 X ( 2 (X 1 2 x X 2 (X 1 2 (4 + 3 X 3 x 2 X 3 (X 1 3 (5 X 2 6 X + 12 X x 3 X 3/2. Writing U, for M,+/2 as in (3.1, V, for M [1], +/2 as in (3.3, an W, for M [1], 2+/2 as in (3.4 an (3.5, we can compute the small values from (3.11 an the equations above; the first few are as follows: U 2, = ( U 4, = 2 3 (2 + 1 ( 2 U 6, = (2 + 1 (36 1 ( 6 2 U 8, = 1 ( (2 + 1 ( U 10, = (2 + 1 ( V 2, = ( V 4, = 3 2 (2 + 1 ( 2 3 V 6, = (25 1 (2 + 1 ( 5 2 V 8, = 7 ( (2 + 1 ( (

18 W 2, = 1 ( 2 W 4, = 2 W 6, = (16 1 ( 4 2 W 8, = ( This leaves only equations (3.2 an (3.6 unresolve. For (3.2 we require M, +t ( 6 2. which arises for t = ( + (2 + 1/ρ 1/2. This only occurs when ρ n 1 as shown in Table 1 (face vertex case an can easily ( be compute in each case using the formula for t M (x. The result when = 2 is 2t ( 2 2+t t, where t = ((2+1/ρ+1/2 = /ρ since ( 4s+2 t is an integer. Expresse in terms of s = /ρ = t 1, this becomes 2s ( 2 2+s+1 s. Note that s = (2+1 ρ/(2ρ, which maes the first factor 4(2+1/(4ρ+2+ρ+1, an this gives simple expressions for particular ρ. For = 4 we compute M 4,3+ /ρ +1 by expaning the formula given above in powers of X an replacing the coefficient of X i 1/2 by ( 4 s+2 B(i, 2, where B is efine in (3.31. This simplifies to M 4,3+s+2 = 2s s 4s 2 ( 2s s ( 2 In the case ρ = 3 the above expression for s shows that the first factor is ( /(4 10. We o not compute this for arbitrary. It is easy to compute using Maple for the particular values of require in this paper (an much larger values as well. Regaring (3.6, this metho only suffices for n = 2 when there is a special term arising from the vertex vertex type, when ψ = 1 an P ω (ρ is ρ r. This is φ(ρm {r/ρ} 2,r/ρ. The binomial is evaluate as ( (r/ρ 1/2 using (3.11 (as for U2 above but with r replace by r/ρ, which is then clearly equal to ( 2 /ρ /ρ. 5 Results from Bouttier-Di Francesco-Guitter Recursions In this section we use the recursions erive in [BoutDG02] to compute the missing A s neee in the formula for A + (r n. We note that these recursions were also erive by Bousquet-Mélou an Jehanne [BousJ06]. 5.1 Recursions for two vertex egrees Specialising the treatment in [BoutDG02] to two vertex egrees, we efine the following generating functions for roote maps with u maring the number of vertices of egree 3. 18

19 2 + 1 an v maring the number of vertices of egree 2h + 1, where an h are both at least 1. (Warning: for convenience, here we use 2h + 1 where we use h in earlier sections. We have no nee of the variable t in [BoutDG02] an set it equal to 1. In that paper, t was use to mar the eges; the number of eges is etermine by the egrees of the vertices. We now efine three generating functions. E(u, v is the generating function for roote maps all of whose vertices have egrees in {2 + 1, 2h + 1}. Γ 1 (u, v is the generating function for roote maps all of whose non-root vertices have egrees in {2 + 1, 2h + 1}, with root vertex egree 1. Γ 2 (u, v is the generating function for roote maps with root vertex egree 1 an another vertex of egree 1 in the root face, an all other vertices of egrees in {2 + 1, 2h + 1}. We now efine S an R to be power series in u an v efine implicitly from the equations where [z i ] enotes coefficient extraction, S = [z 0 ]W R = 1 + [z 1 ]W W = uq 2 + vq 2h, Q = z + S + R/z an z is a ummy variable. Then the equations [BoutDG02, (2.6,(2.1] state that Γ 1 = S [z 2 ]W Γ 2 = R [z 3 ]W ([z 2 ]W 2 E = Γ Γ 2 1. Hence we obtain Γ 1 = S u ( ( 2 2 2i S 2i R +1 i 2i + 1 i i 0 v ( ( 2h 2h 2i S 2i R h+1 i (5.1 2i h + 1 i i 0 Γ 2 = R (S Γ 1 2 u ( ( 2 2 2i 1 S 2i+1 R +1 i 2i i i 0 v ( ( 2h 2h 2i 1 S 2i+1 R h+1 i (5.2 2i + 1 h + 1 i i 0 E = Γ Γ 2 1 (5.3 19

20 where S = u ( ( 2 2 2i 2i i i 0 R = 1 + u ( ( 2 2 2i 1 2i + 1 i 1 i 0 +v ( ( 2h 2h 2i 1 2i + 1 h i 1 i 0 S 2i R i + v i 0 ( 2h 2i S 2i+1 R i ( 2h 2i h i S 2i R h i (5.4 S 2i+1 R h i. (5.5 To use these results in the context of the present paper, note that A(r n/ρ ; m/ρ = [u n/ρ w 0 ]E(= M,+/2 (5.6 A((r/ρ 1 r (n 1/ρ ; m/ρ = [u (n 1/ρ v]e with 2h + 1 = r/ρ, (5.7 A (1 1 r n/2 ; (m + 1/2 = (nr/2[u n/2 v 0 ]Γ 1, (5.8 A (1 2 r n/2 ; m/2 + 1 = (rn/4[u n/2 v 1 ]Γ 1 with h = 0, (5.9 A((r/ρ 2 r (n 2/ρ ; m/ρ = [u (n 2/ρ v 2 ]E with 2h + 1 = r/ρ. ( Close formulae for arbitrary egrees In [BoutDG02] it is shown that the general relations they obtain simplify in the case of cubic (3-regular maps, an eventually prouce the well-nown formula for the number of roote cubic maps. The expansion technique we use in this section emonstrates that the relations also yiel in principle explicit general formula for the numbers of roote maps whose vertices have arbitary egrees, as long as the number of vertices is fixe (in the same spirit as the results in Section 4. We only nee results for two istinct vertex egrees (hence our specialisation in Section 5.1, an at most two vertices of the secon egree, which maes explicit computations easier. Equations (5.4 an (5.5 are use to calculate ([u v 0 ]S, [u v 0 ]R, ([u v 1 ]S, [u v 1 ]R, an ([u v 2 ]S, [u v 2 ]R recursively. In the following we use (x j to enote the rising factorial x(x + 1 (x + j 1 when j 0. We also efine (x 1 = 1/(x 1. We note ( ( ( 2x 2x 2i 2x = ((x + 1 i i 2 /(2i! 2i x i x ( ( ( 2x 2x 2i 1 2x = (x + 1 i i (x i i+1 /(2i + 1! 2i + 1 x i 1 x ( ( ( 2x 2x 2i 2x = (x + 2 i i 1 (x i i+1 /(2i! ( 2x 2i + 1 2i x + 1 i ( 2x 2i 1 x + 1 i = x ( 2x x (x + 2 i i 1 (x i 1 i+2 /(2i + 1!. We also note that to compute the coefficient of u, we only nee to use S j for j 1 on the right han sie of the recursions, an we use the first terms in the expansion 1 ( x (R0 R x = (u + R 1 (uv + R 2 (uv 2 j +. j j=0 20

21 For computational purposes, we set ( 2 z = u, w = ( 2h v, h S = S 0 (z + S 1 (zw + S 2 (zw 2, R = 1 + R 0 (z + R 1 (zw + R 2 (zw 2. It is not ifficult to see from recursions (5.4 an (5.5 that S 0 (z, R 1 (z an S 2 (z contain only o powers of z, an R 0 (z, S 1 (z, R 2 (z contain only even powers of z. Also [z 2+1 ]S 0 (z an [z 2 ]R 0 (z are polynomials of of egree 2 an 2 1, respectively; [z 2 ]S 1 (z an [z 2+1 ]R 1 (z are polynomials of an h with egrees 2 an 2 + 1, respectively; [z 2+1 ]S 2 (z an [z 2 ]R 2 (z are polynomials of an h of egree an 2 + 1, respectively. Proof of Theorem 2: It follows from equations ( that [w 0 ]Γ 1 an [w 1 ]E contain only o powers of z, an [w 1 ]Γ 1, [w 0 ]E, [w 2 ]E contain only even powers of z. Also for 0, [w 0 z 2+1 ]Γ 1 an [w 0 z 2+2 ]E are polynomials of with egree 2 1 an 2, respectively; [w 1 z 2 ]Γ 1, [w 1 z 2+1 ]E an [w 2 z 2 ]E are polynomials of an h with egree 2 1, 2, an 2, respectively. The theorem now follows from Theorem 1 by using ( for the entries in Table 1, an noting that we require 2h an thus h = /i for an integer i. The first few terms of these functions are given below, obtaine using Maple. [w 0 ]Γ 1 = z/( (1/2z 3 + (1/24 2 (25 1z 5 +(1/720 2 ( z 7 +(1/ ( z 9 + [w 1 ]Γ 1 = 1/(h (1/2(2 + hz 2 + (1/24( h h + 12h 2 h 2 z 4 +(1/720( h h h 60h h h + 30h 2 4h h h h h 2 + h 5 z 6 + [w 0 ]E = z 2 + (2/3(2 + 1z 4 + (1/20 2 (2 + 1(36 1z 6 +(1/315 2 (2 + 1( z 8 +(1/ (2 + 1( z 10 +(1/ (2 + 1( z 12 + [w 1 ]E = 2z + (1/3(3 + h + 2(h + 3z 3 +(1/60(h ( h h 2 + h 3 h 2 z 5 + [w 2 ]E = 1 + 2( + h + 1( + hz 2 +(1/12(2h ( h + 24h 2 + 4h 3 h 2 z 4 +(1/90(3 + h + 2( h h h h h h + 120h 4 90h h 2 + 8h 5 12h 4 h 2 + 5h 3 z

22 These expansions can be use in conjunction with ( an Theorem 1 to calculate expressions for A + (r n for all small n. The ones we will nee in particular because they are not obtaine using the other metho are, from (5.10, ( 2 ( /3 A((r/3 2 r 2 ; 4r/3 = (4/9(2 + 1(4 1, (5.11 /3 ( 4 ( 2 2 2h A((r/3 2 r 4 = (1/12(2h ( h + 24h 2 + 4h 3 h 2, h (5.12 an A((r/3 2 r 6 = ( 6 ( 2 2 2h (1/90(3 + h + 2P (, h, h (5.13 P (, h = h h h h h h + 120h 4 90h h 2 + 8h 5 12h 4 h 2 + 5h 3. 6 Final computational results In this section we give general formulae for the number of unroote regular r-valent maps for all o r when n 10. We also compute some actual numbers for n 20. All results are foun counte using (2.1, Tables 1 an 2, an equations (3.1 (3.6 using the appropriate coefficients of M (x an M [1] (x as escribe in Section 4, together with the three particular expressions (5.11, (5.12 an (5.13 for A((r/3 2 r 2 ; 4r/3 an so on. A((r/3 2 r 2 is neee when n = 8 an r > 3 is a multiple of 3, A((r/3 2 r 4 is neee with n = 14 when r = 9 (an hence = 4 an h = 1, an A((r/3 2 r 6 with n = 20 when r = 9 ( = 4, h = 1. The general formulae are the following, for which one shoul recall that δ a b is 1 if 22

23 a b an 0 otherwise, an that = (r 1/2. A + (r 2 = 1 ( 2 1 ( 2 2 /ρ + φ(ρ 2 2(2 + 1 /ρ 1 ρ 2+1 A + (r 4 = 1 ( ( 2 ( ( /3 2 + δ 3 r 3 /3 A + (r 6 = 36 1 ( ( ( 2 ( ( /5 2 + δ 5 r 5 /5 ( 8 A + (r = 2520 ( ( 4 2 ( ( 2 ( 2 ( /3 + δ 3 r (2 + 1(4 1 9r /3 ( ( 6 2 /7 2 +δ 7 r 7 /7 ( A + (r = ( ( ( 2 48 ( ( 2 /3 /3 +δ 3 r ( ( δ 9 r 2 3 ( 2 /9 /9 ( 2 Computing for specific small values of r gives the following table for the number A + (r n of unroote r-regular maps with n vertices.. 10 Table 3: Numbers of unroote regular n-vertex maps, 2 n 20 r A + (r

24 r A + (r r A + (r r A + (r

25 r A + (r r A + (r r A + (r r A + (r r A + (r

26 r A + (r For the reaer s convenience, in Table 4 we represent the same numerical ata for r = 3, 5 an 7 groupe by vertex egrees. Table 4: Numbers of unroote r-regular maps, r = 3, 5, 7 n A + (3 n A + (5 n A + (7 n Acnowlegement We wish to than a referee of an earlier version of this paper, for a suggestion which stimulate further examination of [BoutDG02] an le, though somewhat inirectly, to the use of the recursions therein. References [BenC94] E. A. Bener an E. R. Canfiel, The number of egree restricte roote maps on the sphere, SIAM J. Discrete Math. 7:1 (1994, MR (94:05093 [BouLL02] M. Bousquet, G. Labelle an P. Leroux, Enumeration of planar two-face maps, Discrete Math., 222 (2000, MR (2001c:05073 [BousJ06] M. Bousquet-Mélou an A. Jehanne, Polynomial equations with one catalytic variable, algebraic series an map enumeration, J. Combin. Theory, Ser. B, 96:5 (2006, MR [BoutDG02] J. Bouttier, P. Di Francesco, an E. Guitter, Census of planar maps: from the one-matrix moel solution to a combinatorial proof, Nuclear Physics B 645:3 (2002, MR (2004f:05004 [Gao93] Z. C. Gao, The number of egree restricte maps on general surfaces, Discrete Math. 123 (1993, MR (94j:05008 [GaoR97] Z. C. Gao an M. Rahman, Enumeration of -poles, Annals of Combin. 1:1 (1997, MR (98:

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