Enumeration of m-Ary Cacti

Enumeration of m-ary cacti Miklos Bona Michel Bousquet Gilbert Labelle Pierre Leroux LACIMy Universite du Quebec a Montreal Montreal, Quebec, H3C 3P8 Canada November 3, 1999 Abstract The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classi cation of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti. 1 Introduction A cactus is a connected simple graph in which each edge lies in exactly one elementary cycle. It is equivalent to say that all blocks (2-connected components) of a cactus are edges or elementary cycles, i.e., polygons. An m-gonal cactus (m-cactus for short) is a cactus all of whose polygons are m-gons, for some xed m  2. By convention, a 2-cactus is simply a tree. These graphs were previously called \Husimi trees", and their de nition was given by Harary and Uhlenbeck [12] following a paper by Husimi [13] on the cluster integrals in the theory of condensation in statistical mechanics. See also Riddell [18] and Uhlenbeck and Ford [22]. Their enumeration according to the number of polygons was carried out in [12]. See also Harary and Palmer [11] and [13]. A plane m-cactus is an embedding of an m-cactus into the plane so that every edge is incident with the unbounded region. An m-ary cactus is a plane m-cactus whose vertices are cyclically mcolored 1; 2;


; m counterclockwise within each m-gon. For technical reasons, we also consider  y Present address: Department of Mathematics, University of Florida, Gainesville. With the partial support of FCAR (Quebec) and CRSNG (Canada). 1 a single vertex colored in any one of the m colors to be an m-ary cactus. A quaternary (m = 4) cactus is shown on Figure 1. We de ne the degree of a vertex in a m-ary cactus to be the number of m-gons adjacent to that vertex. Note that it is twice the number of edges adjacent to the given vertex, for m  3. Given an m-ary cactus , let nij denote the number of vertices of color i and degree j of  and set ni = (ni0 ; ni1; ni2 ; : : :). The vertex-degree distribution of  is given by the m  1 matrix N = (nij ),Pwhere 1  i  m and j  0. Note that ni = Pj nij is the number of vertices of color i and n = i ni is the total number of vertices of . The vertex-color distribution of  is de ned to be the vector ~n = (n1 ; n2; : : :; nm ). Also, let p denote the number of polygons in . For the quaternary cactus of Figure 1, the distributions are n1 = (0; 7; 1; 0; 1; 0;


) = 1721 41; n2 = (0; 7; 3; 0; 0; 0;


) = 17 23; n3 = (0; 8; 1; 1; 0; 0;


) = 1821 31; n4 = (0; 9; 2; 0; 0; 0;


) = 19 22; n1 = 9; n2 = 10; n3 = 10; n4 = 11; n = 40; and p = 13; 4 2 3 3 3 1 1 1 2 4 4 2 2 4 1 2 3 4 3 1 2 4 3 1 4 2 4 1 4 4 2 3 1 3 3 2 1 3 2 4 Figure 1: A quaternary cactus. Clearly, for any m-ary cactus with n vertices and p polygons we have X jn j ij = p; for all i, (1) since each polygon contains exactly one vertex of color i, and also n = (m ? 1)p + 1; (2) as one sees easily by induction on p. The goal of this paper is to enumerate various classes of m-ary cacti according to the number n of vertices or p of polygons, to the vertex color distribution ~n = (n1; n2; : : :; nm), and to the vertex-degree distribution N = (nij )1im; j 0 . The species we enumerate are the following: 1. K, the class of all m-ary cacti, 2 2. 3. 4. K , the class of m-ary cacti pointed at vertex of color i (see Figure 5), K3, the class of rooted (i.e., pointed at a polygon) cacti (see Figure 6), Ai, the class of m-ary cacti, planted at a vertex v of color i, i.e., pointed at v with a pair i of half edges attached to v contributing to its degree (see Figure 4), 5. K, the class of asymmetric m-ary cacti, 6. K=s and Ks , the classes of m-ary cacti whose automorphism group is of order s, and a multiple of s, respectively, where s  2. The motivation for the enumeration of m-ary cacti comes from the topological classi cation of polynomials having m critical values. More precisely, two complex polynomials p1 (z ) and p2(z ) are said to be topologically equivalent if there exists two oriention-preserving homeomorphisms of the plane, h1 and h2, such that h1 (p1(z )) = p2 (h2 (z )). Also, a complex number v is called a critical value of the polynomial p(z ) if the equation p(z ) = v has at least one multiple root; all the roots of the equation are then called critical points. Now if a polynomial p(z ) has m critical values fv1; v2; : : :; vm g, we form a simple curvilinear m-gon joining these m critical values fv1; : : :; vmg. Then the preimage under p of this polygon yields an m-ary cactus whose vertexdegree distribution corresponds to the multiplicities of the critical points. For example, Figure 2 shows the cactus corresponding to a degree 8 polynomial p(z ) = c0 + c1z + : : : + c8 z 8 having three critical values v1 ; v2; v3, whose derivative is of the form p0 (z ) = (z ? b)(z ? 1)3(z + 21 )2 (z ? i), where b 2 C is chosen so that p(b) = p(1) = v1 , and p(? 21 ) = v2 , p(i) = v3 . 1 0 –1 1 Figure 2: Cactus associated to a polynomial of degree 8, having three critical values. 3 This is a crucial step in the topological classi cation but the equivalence classes of polynomials are in fact represented by the orbits of m-ary cacti under the action of the braid group. See [7] and [14] for more details. The enumeration of these orbits is an open problem. This work extends to general m  2, previous results of Labelle and Leroux [16] on bicolored plane trees. It also extends results of Goulden and Jackson [9] on the enumeration of rooted mary cacti. They show that rooted m-ary cacti with p polygons, having vertex-degree distribution N = (nij ) are in one-to-one correspondance with decompositions of the circular permutation (1; 2; : : :; p) as the product g1g2


gm of m permutations, where gi has cyclic type (1n 1 2n 2


). In section 2, we state the main functional equations relating the various species of m-ary cacti. We show that all these species can be expressed in terms of planted m-ary cacti which, themselves, satisfy functional equations opening the way to Lagrange inversion. Of particular importance is a Dissymmetry Theorem which relates (ordinary) m-ary cacti to pointed and rooted m-ary cacti. This theorem is closely related to the dissimilarity characteristic theorem for trees, due to Otter and extended to cacti by Harary and Norman [10]. The tree-like structure of a cactus can be emphasized by using an equivalent representation, where a white (= color 0) vertex is placed within each polygon, and joined to the vertices of the polygon, after which the edges of the polygons can be erased. This gives a bijection between m-ary cacti having p polygons and (1 + m)-colored trees having p vertices of color 0, all of degree m. The bijection is illustrated in Figure 3 for a ternary (m = 3) cactus. i i 1 1 2 2 1 3 3 1 2 2 2 3 1 1 3 3 3 Figure 3: Tree-like structure of a ternary cactus In section 3 we establish a particular form of multidimensional Lagrange inversion, which is well adapted to the present situation. It extends the previously known two-dimensional case, in the spirit of Chottin's formulae [5] [6], and use the crucial observation due to Goulden and Jackson [9] that a certain Jacobian matrix reduces to a rank-1 matrix. We then use these results in section 4 to enumerate both labelled and unlabelled m-ary cacti, including the special classes of planted, pointed and rooted m-ary cacti, according to the number of vertices (or of polygons), to their vertex-color distribution and their vertex-degree distribution. We also enumerate m-ary cacti according to the order of their automorphism group, including the asymmetric ones. An alternate method can be used for the enumeration of unlabelled m-ary cacti. It is based on a paper of Liskovets [17] on the enumeration of non rooted planar maps which uses the concept of quotient of a labelled planar map under an automorphism. See Bousquet [2], [3]. 4 In the last section, we present some related enumerative results, concerning labelled free m-ary cacti and unlabelled plane m-gonal cacti having p polygons. We also state a closely related result due to Bousquet-Melou and Schae er [4] on rooted m-ary constellations, having p polygons. Three tables are given in the paper, containing numerical results which illustrate some of the formulas. We have used the species formulation as a helpful unifying framework in this paper. A basic reference for the theory of species is the book [1]. However, the paper remains accessible to anyone with a knowledge of Polya theory applied to graphical enumeration (see [11]). We would like to thank Sacha Zvonkin, for introducing us to the problem of cactus enumeration, and Robert Cori and Gilles Schae er for useful discussions. 2 2.1 Functional equations for m-ary cacti Vertex-color distribution We consider the class K of m-ary cacti as an m-sort species. This means that an m-ary cactus is seen as a structure constructed on an m-tuple of sets (U1 ; U2;


; Um ), the elements of Ui being the (labels for) vertices of color i. Moreover, the relabeling bijections, and in particular, the automorphisms of m-ary cacti are required to preserve the sorts of elements, i.e. the colors. Although we are interested in the enumeration of unlabelled cacti, it is easier to establish the functional equations by giving bijections between labelled structures. If we ensure that these bijections are natural, that is, that they commute with any relabeling, thus de ning isomorphisms of species, then the consequences for both the labelled (exponential) generating function n1 nm X K(x1; x2; : : :; xm) = jK[n1; n2; : : :; nm]j xn1 !





nxm! (3) n1 ;n2 ;:::;nm 1 m and the unlabelled (ordinary) generating function X e K(n1; n2; : : :; nm)xn1 1


xnmm Ke (x1; x2; : : :; xm) = n1 ;n2 ;:::;nm (4) are automatic. Here K[n1; n2; : : :; nm ] denotes the set of m-ary cacti over the multiset [n1 ] + [n2 ]+ : : :+[nm ], with [n] = f1; 2; : : :; ng, and Ke (n1 ; n2; : : :; nm ) denotes the number of unlabelled m-ary cacti having ni vertices of color i, for i = 1; : : :; m. Note that the plane embedding of an m-ary cactus  is completely characterized by the speci cation, for each vertex v of , of a circular permutation on the polygons adjacent to v . We now present functional equations related to the m-sort species Ai , of m-ary cacti, planted at a vertex of color i, Ki , of m-ary cacti, pointed at a vertex of color i, K3, of rooted m-ary cacti. The following notations are used: Xi denotes the species of singletons of sort (or color) i, C denotes the speciesQof (non-empty) circular permutations, L denotes the species of lists (linear orders) and Abi := j 6=i Aj denotes the product of all Aj except Ai . Proposition 1 We have the following isomorphisms of species, for i = 1; : : :; m: Ai = XiL(Abi); (5) i K = Xi(1 + C (Abi)); (6) 3 K = A1A2


Am : (7) 5 Colors: : 1 : 2 : 3 v Figure 4: A planted ternary cactus. The plane embedding of a planted m-ary cactus determines a linear order on the neighboring polygons of the pointed vertex. If this vertex, say of color 1, is removed, each of these adjacent polygons can be simply decomposed into the product of m ? 1 planted m-ary cacti with roots of color 2; 3; : : :; m. Since this data completely speci es the planted cactus, we have equation (5). See Figure 4 for an illustration of the equation A1 = X1 L(A2A3 ) in the ternary case. Equation (6) is similar to (5) except that for pointed cacti the polygons adjacent to the pointed vertex can freely rotate around it. Figure 5 illustrates the equation K3 = X3(1 + C (A1A2 )). Equation (7) is immediate; see Figure 6. 2 Remark that equations (5) and (7) are essentially due to Goulden and Jackson [9]. Proof: v Figure 5: A ternary cactus pointed at vertex v . Recall that in a connected graph g , a vertex x belongs to the center of g if the maximal distance from x to any other vertex is minimal. In particular, if g is a cactus, then it is easy to see that the center of g is either a single vertex or a polygon. Now let  be an m-ary cactus. In this case we de ne the center in a slightly di erent way: if the previous de nition yields a vertex as the center of , then we leave this de nition unchanged. If the previous de nition yields a 6 polygon p as the center of , then we take the color-1 vertex of p to be the center of . So now the center of an m-ary cactus is always a vertex. Figure 6: A rooted ternary cactus. Theorem 2 Dissymmetry theorem for m-ary cacti. There is an isomorphism of species (8) K1 + K2 +


+ Km = K + (m ? 1)K3: c z z center y y t x x Figure 7: K1 + K2 + K3 = K + 2K3 (m = 3) Proof: For clarity, we prove the theorem for m = 3, that is we establish an isomorphism K1 + K2 + K3 = K + 2K3, the proof for general m being analogous. The left hand side corresponds to cacti which have been pointed at a vertex, of color 1, 2, or 3. The rst term of the right hand side corresponds to cacti which have been pointed in a canonical way, at their center. So what remains to construct is a natural bijection from triangular cacti pointed not in their center onto two cases of A1 A2 A3 -structures. Suppose that a ternary cactus  has been pointed at a vertex x of color 1 which is di erent from the center c of  (see Figure 7). Let the shortest path from x to c start with the edge 7 e = fx; yg, and let t be the unique triangle containing e. Then we cut the three edges of t and thus separate the cactus into three smaller cacti which are planted in a vertex of color 1, 2 and 3 respectively. We thus obtain an A1 A2 A3 -structure. It is easy to see that we could have obtained this structure in another way. Indeed, if the vertices of t are x; y and z , then pointing the cactus at z would have given the same decomposition. So this operation does de ne a map into 2A1 A2 A3 . To see that the algorithm is reversible, take any 3-tuple of cacti which are planted in vertices x, y and z of color 1, 2, and 3 respectively. Join x, y and z by a triangle to get a cactus, and look for its center c. If c comes from the component of x, then we can point either y or z in the cactus, if c comes from the component of y , then we can point either x or z and nally, if c comes from the component of z , then we can point either x or y . It is then a simple matter to number each of these cases in order to make the correspondence bijective, completing the proof. 2 Corollary 3 The species K of m-ary cacti can be written as K = = Xm K ? (m ? 1)K3 i Xm Xi(1 + C(Abi)) ? (m ? 1) Ym Ai: i =1 i=1 i=1 (9) 2 The consequences for the labelled and unlabelled generating functions then follow from general principles. For i = 1; : : :; m, we have, with x = (x1; x2; : : :; xm), (10) Ai(x) = xi 1 ; 1 ? Ai (x) Ai(x) = xi 1 ; (11) 1 ? Ai (x) b fb f f fb ce from which it follows that Ai (x) = Ai (x) since Ai = Ai . This expresses the fact that planted cacti are asymmetric structures. Moreover, K (x) = xi(1 + log 1 ); (12) 1 ? Ai (x) (13) K (x) = xi(1 + (dd) log 1 d ); 1 ? Ai (x ) d1 where xd := (xd1 ; xd2 ; : : :; xdm) and  is the Euler function. We also have b i X g i b g eK(x) = Xm Kg (x) ? (m ? 1)K3(x): K3(x) = K3(x) = A1 (x)


Am(x) and nally, i i=1 8 (14) (15) 2.2 Vertex-degree distribution In order to enumerate m-ary cacti Q according to their degree distributions, we introduce weights in the form of monomials w() = i;j rijnij with i = 1; : : :; m and j  0, for a cactus  having vertex-degree distribution N = (nij ). In other words, the variable rij acts as a counter for (or marks) vertices of color i and degree j . We also use the notation ri to denote the sequence (ri0; ri1; : : :). We denote by Kw ; Kw3, and Kwi the corresponding species of m-ary cacti, weighted in this manner. We denote by Ai;r the species of planted (at a vertex of color i) m-ary cacti similarly weighted by degree. The functional equations (5){(9) can then be extended as follows: Q where Abi;r = j 6=i Ai;r ; Ai;r = Xi(ri;1 + ri;2Ab2i;r + ri;3Ab3i;r + : : :) (16) Kwi = Xi(ri;0 + ri;1C1(Abi;r) + ri;2C2(Abi;r) + : : :); (17) where Ck denotes the species of circular permutations of length k, m Y 3 Kw = Ai;r; i=1 and Kw = m X i=1 (18) Kwi ? (m ? 1)Kw3: (19) The important point here is that the weights behave multiplicatively, with respect to the operations of product and partitional composition. The consequences for the labelled and unlabelled generating functions are as follows: Ai;r(x) = xi(ri;1 + ri;2Abi;r(x) + ri;3Ab2i;r(x) + : : :); Agi;r(x) = Ai;r(x); X Kwi (x) = xi(ri;0 + rhi;h Abhi;r(x)); Kgwi (x) = xi(ri;0 + h1 X ri;h X h1 h djh d (d)Abh=d i;rd (x )); (20) (21) (22) (23) d g, for i = 1; : : :; m, j  0. We also have where rd denotes the set of variables fri;j Kw3(x) = Kgw3(x) = and nally, m Y i=1 Ai;r(x); X Kgw (x) = Kgwi (x) ? (m ? 1)Kw3(x): m i=1 9 (24) (25) 2.3 One-sort m-ary cacti If neither the vertex-color nor the vertex-degree distribution are desired, but only the number of vertices or, equivalently, of polygons, then the enumeration is easier to carry out since one dimensional Lagrange inversion will suce. Indeed, we can consider the various species of mary cacti introduced earlier as one-sort species, as Figure 1 suggests. This means that the underlying set (of vertex labels) is independant of the colors and that the relabellings can be arbitrary, altough isomorphisms are still required to preserve colors. We use the same letters K; Ki ; K3; Ai to denote these one-sort species. Equations (6){(8) are still valid in this context, with the following simpli cations: rst, all singleton species Xi should be replaced by X ; second, the addition of 1 modulo m to the colors induces isomorphisms of species A1  = :::  = Am , = A2      m  2 . : : : K K and we write A for this common species, and also K 1  = = = Equation (5) then simpli es to A = XL(Am?1) = 1 ? X Am?1 ; (26) K3 = Am = A ? X; (27) Ki = X (1 + C (Am?1)): (28) K = K ? (m ? 1)K3 = mX (1 + C (Am?1 )) ? (m ? 1)(A ? X ); (29) which implies A = X + Am . Moreover, equation (7) reduces to while (6) reduces to Finally, the dissymetry theorem for m-ary cacti takes the form where K denotes the one-sort species of pointed at (any color) m-ary cacti. 3 Multidimensional Lagrange inversion techniques In this section we establish a special form of multidimensional Lagrange inversion, which can be directly applied to m-ary cacti. First recall the standard form, due to Good, (see Theorem 1.2.9, 1 of [8] or the equivalent formula (28b) of [1]). Theorem 4 Good's Lagrange Inversion Formula. Let A1; A2;


; Am be formal power series in the variables x1; x2;


; xm such that the relations Ai = xi Ri(A1 ; A2;


; Am) are satis ed for all i = 1;


; m. Then for any formal power series F (t1 ; t2;


; tm ) we have: [xn1 1


xnmm ]F (A1 (x);


; Am(x)) = [tn1 1


tnmm ]F (t)jK (t)jRn1 1 (t)


Rnmm (t); (30) where t = (t1 ; t2;


; tm ) and K (t) is the m  m matrix whose (i; j )-th entry is K (t)ij = ij ? Rt(jt)
@R@ti(t) : i 10 j (31) 2 There is a particularly simple two-dimensional case of this formula, the alternating case, which we call the Chottin formula. In the papers [5] [6], Chottin worked extensively on the two-dimensional Lagrange inversion and its combinatorial proof. Theorem 5 Chottin Formula. Let A(x; y) and B(x; y) be two formal power series satisfying the relations A = x(B ) and B = y (A), where (t) and (s) are given formal power series. Then, for any non negative integers and we have: [xn y m ]A B = (1 ? (n ? )(m ? ) )[sn? tm? ]n (t) m (s); n  1; m  1: (32) nm 2 We extend this result into m dimensions. Theorem 6 Generalized Chottin formula. Let A1 ; A2; : : :; Am be formal power series in the variables x1 ; x2; : : :; xm such that for i = 1; : : :; m, the relationsQAi = xi i (Abi ) are satis ed, where the i are given formal power series of one variable, and Abi = P ; : : :; nm j 6=i Aj . Also let n1P m be integers  1 and let 1 ; : : :; m be nonnegative integers. Set n = i=1 ni and = m i=1 i . Suppose that the following coherence conditions are satis ed, ni  i ; n? = m?1 is an integer; and set i = ? ni + i . Then [xn1 1


xnmm ]A1 1


Amm = D
[s1 1


smm ]n1 1 (s1 )


nmm (sm ); where m X Ym j Y(1 + i ): D = (1 + i ) ? i=1 ni j =1 nj i6=j ni (33) (34) Proof: We use Theorem 4 with Ri(t1; : : :; tm) = i(tbi ), where bti = Qj6=i tj . We take advantage of some useful observations made by Goulden and Jackson in [9] to compute the determinant jK (t)j. Indeed, for i = 1; 2;


; m, we have i tj @R @t = 0; as Ri(t) does not depend on ti , and for j 6= i, i i b 0 b tj @R @t = ti i (ti ) j which is independent of j . We set i (bti ) = tbi 0i (bti ) and write i = i (bti ); i = i(bti ). The de nition of K (t) then yields, after routine transformations, Qmi=1(i + i) jK (t)j = Qm 
jij ? +i  j: i i i=1 i i and note that the rank of M is 1 since all its columns are equal. So, by Let Mij = ? i + i the Sherman-Morrison formula [20] we have jI + M j = 1 + trace(M ). Therefore, the previous equation yields Qm (i + i) m X i ): (35) jK (t)j = i=1Qm 
(1 ? + i=1 i i=1 i 11 i It follows from the Lagrange inversion formula (30) that Y [xn1 1


xnmm ]A1 1


Amm = [tn1 1


tnmm ]t1 1


tmm
jK (t)j ni i m i=1 = [t1n1 ? 1 Ym


tnm ? m ]( ni?1 ( + m Now let us de ne the coecients ci; i by ni i (bti ) = i=1 m X i=1 i i + i ): (36) X c bt i ; i; i i (37) i 0 which implies, by the de nition of i that ini ?1 (bti ) i (bti ) = P i i i ))(1 ? P X i c bt i : i; i i n i i 0 (38) m 1 n? Recall that n = mi=1 ni and = mi=1 i . Then t1 1 1


tnmm ? m = tb1


tc m if and only if P ? i = ni ? i for all i,where = mi=1 i. Summing these equations for i = 1; : : :; m yields (m ? 1) = n ? and i = ? ni + i . We then conclude that (36) equals [bt1 1


btmm ]( Ym ni ?1( + i=1 i i ))
(1 ? i m X i=1 i ) = D
[s 1


s m ]n1 (s )


nm (s ); m m m 1 1 1 + i i 2 where D is given by (34), completing the proof. The following special cases are particularly useful: 1. 1 = 2 =


= m = 1, with the condition that (n ? 1)=(Q m ? 1) = p is a positive integer. m ? 1 Then we nd that = p ? 1, i = p ? ni and D = p = mi=1 ni , and we have m?1 [xn1 1


xnmm ]A1


Am = Qpm n
[sp1?n1


spm?nm ]n1 1 (s1)


nmm (sm ): i=1 i 2. (39) P = 0, 2 =


= m = k  1, with the condition that ( i ai )=(m ? 1) = q is an integer. Then we ndQthat = q ? k, 1 = q ? a1 ? k, i = q ? ai , for i = 2;


m, and that D = q m?2 k= i6=1 ai , and we have, writing Ab1 = A2A3


Am , 1 m?2 k [xa11


xamm ]Abk1 (x) = Qq q?a1 ?k sq?a2


sq?am ]a1 (s )


am (s ): 1 m m m 1 2
[s1 i6=1 ai (40) P 3. Under the condition that ( i ai )=(m ? 1) = q is an integer, it follows from (40) that for any formal power series F (s) we have [xa11


xamm ]F (Ab1 ) = Qq m?2 q?a1 ?1 sq?a2


sq?am ]F 0 (s )a1 (s )


am (s ): 1 1 1 m m m 2
[s1 i6=1 ai 12 (41) 4 Enumeration of m-ary cacti 4.1 Coherence conditions As observed in the introduction, there are some coherence conditions on the statistics of an m-ary cactus. We now state necessary and sucient conditions for the existence of an m-ary cactus. The rst one concerns the relationship between the number of vertices and the number of polygons. It is easily proved by induction on p. Lemma 7 There exists an m-ary cactus having n vertices and p polygons if and only if n = p(m ? 1) + 1: 2 Lemma 8 Let ~n = (n1; n2;


; nm) be a vector of nonnegative integers and set n = Pi ni . There exists an m-ary cactus having n vertices, p polygons and vertex-color distribution ~n if and only if 1. p = (n ? 1)=(m ? 1) is an integer, 2. p  1 ) ni  p, for i = 1;


; m. Proof: The conditions are clearly necessary. Suciency is proved by induction on p. If p = 0, then n = 1, and we have a 1-vertex cactus. If p  1, then all components of ~n are strictly positive since otherwise, supposing for example that n1 = 0, we nd m X n= n i=2 i  (m ? 1)p = n ? 1; a contradiction. Hence we have ni  1, for all i. If p = 1, then ni = 1 for all i, and we have a cactus with a single polygon. If p > 1, we must have ni < p for some i, since otherwise n = mp and n = p(m ? 1) + 1 leads to a contradiction. Assume, say, nm < p and de ne a new vector ~n0 by n0m = nm and n0i = ni ? 1 for i = 1;


; m ? 1: This vector ~n0 satis es the conditions 1 and 2 with (n0 ? 1)=(m ? 1) = p ? 1 and we can apply the induction hypothesis to construct a cactus with vertex distribution ~n0 . It suces then to add a new polygon to this cactus, attached to any existing vertex of color m to obtain a cactus with vertex-color distribution equal to ~n. 2 Observe that when conditions 1 and 2 are satis ed, p  1 ) ni  1 for all i. Lemma P 9 Let N = (nij )1im;j0 be an m  1 matrix of non negative integers, and set n = ij nij . There exists an m-ary cactus having n vertices and p polygons and whose vertexdegree distribution is given by the matrix N if and only if 1. p = (n ? 1)=(m ? 1) is an integer, 2. Pj jnij = p, for all i, 3. p  1 ) ni0 = 0, for all i. 13 Proof: These conditions are clearly necessary. Suciency is again proved by induction on p. If p = 0, then n = 1 and we have a one vertex cactus. If p  1, then we can prove that for P all i, except possibly one, ni  1. Indeed conditions 2 and 3 imply that ni = j nij  p. Then,Pif ni = 0 for some i, we have ni  p=2. If this occurs for two or more values of i, then n = i ni  (m ? 1)p = n ? 1, a contradiction. If p = 1, then ni = 1 for all i and we have a one polygon cactus. If p > 1, then either one ni = 0, say nm = 0, or all ni are  1. In the rst case there must be some j  2 with nmj  1; in the second case, there must exist somePi, say i = m, and some j  2, with nmj  1 since otherwise ni = ni = p for all i and n = ni = mp = (m ? 1)p + p, a contradiction. In either case we set n0i = ni ? 1 for i = 6 m, n0mj = nmj ? 1, n0m;j? = nm;j? + 1 and n0ij = nij for other i; j . Then the matrix N 0 = (n0ij ) satis es the conditions of the lemma with p0 = p ? 1 and we can apply the induction hypothesis 1 1 1 1 1 1 1 1 1 1 1 to construct a cactus with vertex-degree distribution N 0. It remains then to add a new polygon to this cactus, attached to any existing vertex of color m and degree j ? 1 to obtain a cactus with vertex-degree distribution N . 2 m-ary cacti As observed earlier, the species K3 of rooted m-ary cacti is asymmetric. It follows that labelled 4.2 Rooted or labelled m-ary cacti and rooted m-ary cacti are closely related. For example, in the one-sort case, we have (42) pKn = Kn3 = n!Ke n3 where Kn and Kn3 denote the number of m-ary cacti and rooted m-ary cacti, respectively, having n labelled vertices, and Ke n3 denotes the number of unlabelled m-ary cacti with n vertices, and where p is the number of polygons. Theorem 10 Let p be a positive integer and set n = p(m ? 1) + 1. Then the numbers Ke n3, of rooted (unlabelled) m-ary cacti, and Kn , of labelled m-ary cacti, having n vertices (and p polygons), are given by ! eKn3 = 1 mp (43) n p and ! mp ( n ? 1)! : Kn = p p (44) Proof: It follows from (26) and (27) that the one-sort species A and K3 of planted and rooted m-ary cacti respectively satisfy A(x) = x=(1 ? Am? (x)) and Ke 3(x) = K3(x) = A(x) ? x. The 1 result follows easily from Lagrange inversion since Ke n3 = [xn](A(x) ? x) = 1 [tn?1 ](1 ? tm?1 )?n n = n1 [t(m?1)p ](1 ? tm?1 )?((m?1)p+1) ! mp 1 : = n p 14 2 The second result then follows from (42). Remark 11 Formula (43) also represents the number of (unlabelled) m-ary ordered rooted trees having p internal vertices and n leaves. A direct bijection can be given between rooted m-ary cacti and m-ary ordered rooted trees, which also explains the functional equation A = X + Am . See [2] and [3]. Suppose now that a vector ~n = (n1 ; n2; : : :; nm) satis es the conditions of Lemma 8. Let K~n denote the number of m-ary cacti over the multiset of vertices ([n1]; [n2]; : : :; [nm]), that is, of labelled cacti with vertex-color distribution ~n. Similarly let K~n3 denote the number of labelled rooted m-ary cacti with vertex distribution ~n. Then we have m Y (45) p
K~n = K~n3 = ( ni !)Ke ~n3; i=1 where Ke ~n3 is the number of unlabelled rooted cacti with vertex-color distribution ~n. Theorem 12 Let ~n = (n1; n2; : : :; nm) be a vector of nonnegative integers satisfying the coherence conditions of Lemma 8, with p  1. Then the number of unlabelled rooted m-ary cacti having vertex distribution ~n is given by ! m eK~n3 = 1 Y p : (46) n p i=1 i Proof: Recall that Ke 3(x) = K3(x) = A1(x)A2(x)


Am(x) and that the Ai (x) satisfy func- tional equation (10). Hence we can use the special case 1 of the Generalized Chottin formula, that is, formula (39), with i (s) = L(s) = 1=(1 ? s) for all i. Hence we nd that Ke~n3 = [xn1 1


xnmm ]A1(x)


Am (x) m m?1 Y = Qpm n [spi ?ni ](1 ? si )?ni i=1 i i=1 m p?1! m?1 Y p ; = Qm n i=1 i i=1 ni ? 1 2 which implies (46). Putting together equations (45){(46) yields the following: Corollary 13 If the conditions of Lemma 8 are satis ed, the number of labelled m-ary cacti with vertex-color distribution ~n = (n1 ; n2;


; nm ) is given by m Y K~n = pm?2 (p ? ni + 1)<ni?1> ; i=1 (47) where x<k> denotes the rising factorial x(x + 1)


(x + k ? 1). 2 ?1> <n1 ?1> Remark 14 This extends to general m  2 the formula n<n n2 for the number of 1 labelled plane bicolored trees with vertex-color distribution (n1 ; n2) (see formula (2.7) of [16]). 15 To nd the number KN of labelled m-ary cacti having vertex-degree distribution N = (nij ), where i = 1;


; m and j  0, a similar approach can be followed. As for the vertex-color distribution, we have p
KN = n1 !


nm !Ke N3 ; (48) P where ni = j nij and Ke N3 denotes the number of (unlabelled) rooted m-ary cacti having vertexdegree distribution N . Recall that ni = (ni0 ; ni1; ni2; : : :) is the degree distribution for vertices of color i. The following result, due to Goulden and Jackson [9], expresses the number KgN3 in ?n
terms of the multinomial coecients n . i i Theorem 15 [9] Let N = (nij )1im;j0 be an mP1 matrix of non negative integers satisfying the coherence conditions of Lemma 9, with n = ij nij and p = (n ? 1)=(m ? 1)  1. Then the number of rooted m-ary cacti having nij vertices of color i and degree j , is given by ! m n m?1 Y i : KN = Qpm n i=1 i i=1 ni e3 (49) Proof: Recall that Kw3(x) = Ke w3(x) = Qmi=1 Ai;r(x) and also recall equations (20). Again, we use the generalized Chottin formula (39), with i (s) = r (s) := ri1 + ri2s + ri3s2 +


Then we have (50) i m?1 m Y p [ rn ][ sp?n ] n (s ); Ke N3 = [ rijn ][ xni ]Kw3(x) = Q i i i ij n Y i;j ij Y i Y Y ij i i i;j i i i i=1 i (51) 2 which implies (49). Corollary 16 The number KN of labelled m-ary cacti having vertex-degree distribution N , assuming that the conditions of Lemma 9 are satis ed, with p  1, is given by KN = pm?2 m Y i=1 ! (ni ? 1)! nni : i (52) 2 Remark 17 It is well-known that the number of ways to label an unlabelled structure  over an underlying multiset [n1 ; n2; : : :; nm ] is n1 !n2 !


nm != j Aut() j, where Aut() denotes the (color-preserving) automorphism group of . It follows that X 2! : : :nm ! (53) KN = n1j!nAut( ) j  where the sum is taken over all unlabelled m-ary cacti  with vertex-degree distribution N . It also follows that X 1 = 1 Ke 3 : (54) p N  j Aut() j This formula can be used, as in [7], to check that all unlabelled cacti with a given degree distribution have been found. 16 m-ary cacti (unlabelled) Recall that K = K(X ) denotes the one-sort species of m-ary cacti which are pointed at a 4.3 Pointed vertex of any color. We have Ke (x) = Ke 1 (x) + : : : + Ke m (x) = mKe 1 (x): Theorem 18 Let p be a positive integer and set n = p(m ? 1) + 1. Then the number Ke n of pointed m-ary cacti having n vertices (and p polygons), is given by ! eKn = 1 X (d) pm=d ; p djp p=d (55) where  is the Euler function. Proof: We have Ke n = mKfn1 and K1 = X (1 + C (Am?1)). By Lagrange inversion, we nd for p  1; n  m, Kfn1 = [xn](Kfn1 (x) ? x) X (d) 1 log m 1 ? A ?1 (xd ) d1 d X (d) n?1 1 d ] log [ x m 1 ? A ?1 (x) djn?1 d X (d) n?d?1 m?2 m?1 )? n+dd?1 d ]t (1 ? t ( m ? 1)[ t djn?1 n ? 1 X (d) p?d n+d?1 [t d ](1 ? t)? d p djp ! X (d) pm d ?1 p ?1 d djp p ! 1 X (d) pm=d ; mp djp p=d = [xn ]x = = = = = 2 which completes the proof. We now wish to compute the numbers Ke ~ni and Ke Ni of (unlabelled) m-ary cacti pointed at a vertex of color i, with vertex-color distribution ~n and vertex-degree distribution N , respectively. For symmetry reasons, it is sucient to consider the case i = 1 since we have Ke~ni = Ke 1i?1~n and Ke Ni = Ke 1i?1N (56) where  denotes a cyclic shift of the components of ~n or of the rows of N , i.e. (~n)i = ni+1 and (N )ij = ni+1;j ; the sum i + 1 being taken modulo m. We introduce the following notations: ~ek = (ki ); i = 1; : : :; m; eh = (jh )j0 ; Er;s = (ir
js )1im; j0 : 17 (57) (58) Theorem 19 Let ~n = (n ; n ; : : :; nm) bePa vector of non negative integers satisfying the coherence conditions of Lemma 8, with n = i ni and p = (n ? 1)=(m ? 1)  1. Then the number 1 2 of m-ary cacti pointed at a vertex of color 1 and having vertex-color distribution ~n is given by Ke~n1 = ! !Y p=d ; p=d p ? n1 + 1 X (d) (n1 ? 1)=d i6=1 ni =d p2 d (59) where the sum is taken over all d such that d divides p and all components of ~n ? e~1. Proof: Recall equation (13), with i = 1. In what follows we use the special case 3 of the generalized Chottin formula, i.e. (41), with F (s) = log ?s and a = (n ? 1)=d, a = n =d,: : :, 1 1 1 1 2 am = nm =d, so that q = p=d. We nd Ke~n1 = [xn1 1


xnmm ](Ke i(x) ? x1) X 1 = [x~n ]x1 (dd) log 1 ? Ab1(xd ) d1 X 1 = [x~n?~e1 ] (dd) log b 1 ? A1 (xd) d1 X (d) (~n?~e1 )=d 1 [x ] log = d 1 ? Ab1 (x) dj~n?~e1 m q?ni m?2 p?n1 ?d+1 X n +d?1 Y = (d) Qpm n
[s1 d ]( 1 ?1 s ) 1 d [si d ]( 1 ?1 s )ni =d 1 i i=2 i i=2 dj(p;~n?~e1 ) ! ! m (p ? d)=d m?2 X Y = (d) Qpm n ((np ??d1))=d =d i=2 (ni ? d)=d ; 1 i=2 i dj(p;~n?~e1 ) 2 2 which is equivalent to (59). Theorem 20 Let N = (nij ) be an mP1 matrix of non negative integers satisfying the coherence conditions of Lemma 9, with n = ij nij and p = (n ? 1)(m ? 1)  1. Then the number of m-ary cacti pointed at a vertex of color 1 and having n vertices of color i and degree j , is given by !Y ! m?2 X n =d p ( n ? 1) =d i 1  ; (60) Ke N1 = Q n (d) (n ? e )=d n =d 1 h i6=1 i h;d i6=1 where the sum is taken over all ordered pairs (h; d) such that n1h 6= 0 and d divides h; p and all components of n1 ? eh and of ni with i  2. Proof: Recall that ni = Pj nij and ni = (nij )j . We will use the special case 2 of the generalized Chottin formula, i.e. (40), with i (s) = rdi (s) (see (50)), k = h=d, a = (n ? 1)=d, a = n =d, : : :, am = nm =d, for which q = p=d, and Ai (x) = Ai;rd (x). Since p  1, we have, 0 1 2 2 using formula (23) with i = 1, Y Y Ke N1 = [ rijnij ][ xni i ](Ke w1(x) ? r1;0x1) ij i 18 1 = [rN ][xn] = X1 h h X x r ;h X 1 1 h h [rN ?E1;h ] X pm?2 = Q i6=1 ni h;d pm?2 X = Q i6=1 ni h;d pm?2 X djh X (d)Abh=d (xd ) 1;rd dj(h;~n?~e1 ) (d)[x(1n1?1)=dx2n2 =d


xnmm =d]Abh=d (x) 1;rd Y p?ni ni p?n1 ?h+1 n1 ?1 d ] rd1 d (s1 ) [si d ] rdd (si ) i i6=1 (d)[rN ?E1;h ][s1 n1?eh (d)[r1 d Y ni p?ni ni p?n1 ?h+1 n1 ?1 d ] rd1 d (s1) [ id ][si d ] rdi (si ) i6=1 ][s1 r !Y m n =d! ( n i 1 ? 1)=d = Q n (d) (n ? e )=d ; (61) 1 h i6=1 i d;h i=2 ni =d where the summation is taken over all ordered pairs (h; d) such that n1;h 6= 0 and d divides h, p, n1 ? eh , and all ni with i  2. 2 4.4 m-ary cacti (unlabelled) In order to enumerate unlabelled and unrooted m-ary cacti, two methods can be used. The rst one uses the dissymetry theorem for cacti (see Theorem 2) which expresses the species of m-ary cacti in terms of pointed and of rooted cacti; see below. The second is Liskovets' method for the enumeration of unlabelled planar maps [17]. It uses the Cauchy-Frobenius theorem (alias Burnside's Lemma) and the concept of quotient of a planar map under an automorphism; see [2] and [3] for the application of Liskovet's method to the enumeration of m-ary cacti. Theorem 21 Let p be a positive integer and set n = p(m ? 1) + 1. Then the number Ke n of (unlabelled) m-ary cacti having n vertices (and p polygons), is given by 0 ! X !1 md C Ke n = p1 B @ n1 mp p + (p=d) d A; (62) djp d<p where  is the Euler function. Proof: Using the dissymmetry formula (29) for one-sort m-ary cacti, we nd Kfn = Kfn ? (m ? 1)Ke n3 (63) 2 and the result follows from (43) and (55). See Table 1 for some numerical values of Ke n . Theorem 22 Let ~n = (n ; n ; : : :; nm) bePa vector of non negative integers satisfying the coherence conditions of Lemma 8, with n = i ni and p = (n ? 1)=(m ? 1)  1. Then the number Ke~n of (unlabelled) m-ary cacti having vertex-color distribution ~n is given by 0m ! !Y !1 X Y p=d A; (64) Ke ~n = p1 @ np + (d)(p ? ni + 1) (n p=d ? 1) =d n i i i i;d j 6 i j =d 1 2 2 =1 = 19 where the sum is taken over all pairs (i; d) such that 1  i  m; d > 1, d divides p and all components of ~n ? e~i . Proof: Using the dissymmetry formula (15), we have m X Ke ~n = Ke~n ? (m ? 1)Ke ~n3: (65) i i=1 2 The result follows from (46), (56) and (59). See Table 2 for some numerical values of Ke ~n . Theorem 23 Let N = (nij ) be an P m1 matrix of non negative integers satisfying the coherence conditions of Lemma 9, with n = ij nij and p = (n ? 1)=(m ? 1)  1. Then the number Ke N of (unlabelled) m-ary cacti having nij vertices of color i and degree j , is given by 0 m eKN = pm?2 @ Y 1 i=1 ni 1 ! ! ! ni + X Q(d) (ni ? 1)=d Y n` =d A; ni i;h;d `6=i n` (ni ? eh )=d `6=i n` =d (66) where the sum is taken over all triplets (i; h; d) such that nih 6= 0; d > 1; and d divides h; p and all entries of the matrix N ? Eih . Proof: The dissymmetry formula (19) gives X Ke N = Ke N ? (m ? 1)Ke N3 : m (67) i i=1 2 The result follows from (49), (56) and (60). See Table 3 for some numerical values of Ke N . 4.5 Unlabelled m-ary cacti according to their automorphisms We rst consider asymmetric m-ary cacti, that is, cacti whose automorphism group is reduced have already observed to the identity. Let K denote the species of asymmetric m-ary cacti. We 3 3 that the species K of rooted m-ary cacti is asymmetric i.e. that K = K3. The dissymetry formulas (29) and (8), yields, in the one-sort case, K = K ? (m ? 1)K3; and in the m-sort case, K= m X i=1 (68) K ? (m ? 1)K3: (69) i ci)), the enumeration of (unlabelled) K -structures uses the asymmetry Since K = X (1 + C (A index series ?C of the species C of circular permutations, instead of the cycle index series ZC for the enumeration of unlabelled cacti (see [15], [1]), where X (70) ?C (x1 ; x2; : : :) = (d) log 1 ; d 1 ? x d d1 i i 20 compared to the cycle index series X (d) 1 ; log (71) 1 ? xd d1 d where  is the Mobius function. It follows that the enumeration formulas for asymmetric m-ary cacti will be very similar to those of unlabelled cacti. In fact it suces to replace  by  in the formulas of the previous section. Hence we have the following theorem. Theorem 24 Assume that the coherence conditions of Lemmas 7, 8 and 9 are satis ed, with p  1. Then the corresponding enumerative formulas for (unlabelled) asymmetric m-ary cacti are as follows: ZC (x1; x2; : : :) = 0 ! X !1 md C; Kn = p1 B@ n1 mp +  ( p=d ) p d A dp 0 m ! d<p !1 !Y X Y p=d A; K~n = p12 @ np + (d)(p ? ni + 1) (n p=d n ? 1) =d i i=1 i j 6=i j =d i;d 0m !1 !Y ! X Y n` =d A KN = pm?2 @ n1 nni + Q(d)n ((nni??e1))=d=d n =d ; (72) j i=1 i i `6=i ` i;h;d i h `6=i ` where the summation ranges of (73) and (74) are the same as for (64) and (66). (73) (74) 2 We now consider m-ary cacti admitting at least one non trivial automorphism. Since automorphisms are required to preserve colors, the only possibilities are rotations around a central vertex. See Figure 8. Observe that the order of such an automorphism must divide the number p of polygons. Let s  2 be an integer. Let K=s ; and Ks ; denote the species of m-ary cacti whose automorphism groups (necessarily cyclic) are of order s, and a multiple of s, respectively. Then, following the notations of [16], section 3, we have K=s = Ks = m X i=1 m X i=1 XiC=s (Abi ); (75) XiCs (Abi ): (76) We can determine the unlabelled generating series Ke s (x) and Ke =s (x) by formulas (3.2) and (3.3) of [16], essentially due to Stockmeyer. See [1], Exercise 4.4.16, and [21]. Extracting coecients in these series is similar to the computations of subsection 4.3. We nd the following. Theorem 25 Let s  2 be an integer and assume that the coherence conditions of Lemmas 7, 8 and 9 are satis ed, with p a multiple of s. The corresponding enumerative formulas for (unlabelled) m-ary cacti whose automorphism groups are of order s, and a multiple of s, respectively, are as follows: ! eK=s;n = s X (d) pm=sd p=sd p dj ps 21 (77) Figure 8: A ternary cactus with a symmetry of order 3. and ! X pm=sd s Ke s;n = p (d) p=sd ; dj p (78) !Y ! m s(p ? n + 1) X X p=d p=d i (d=s) (n ? 1)=d ; Ke =s;~n = p2 i i=1 d j 6=i nj =d (79) !Y ! m s(p ? n + 1) X X p=d p=d i Ke s;~n = (d=s) (n ? 1)=d ; p2 i i=1 d j 6=i nj =d (80) s and the second summations being taken over all integers d such that sjd and d divides p and all components of ~n ? ~ei ; ! ! m m?2 eK=s;N = X Qp s X (d=s) (ni ? 1)=d Y nj =d ; (ni ? eh )=d j 6=i nj =d i=1 j 6=i nj h;d (81) and ! ! m m?2 eKs;N = X Qp s X (d=s) (n1 ? 1)=d Y nj =d ; (82) (ni ? eh )=d j 6=i nj =d i=1 j 6=i nj h;d the second sommations being taken over all pairs of integers h; d  1 such that nih 6= 0; s j d, and d divides h and all entries in N ? Eih . 2 22 m=2 p n=p(m?1)+1 0 1 2 3 4 5 6 7 8 9 10 11 12 p 0 1 2 3 4 5 6 7 8 9 10 11 12 p 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 Ke n n K en Kn H en n=p(m?1)+1 1 1 1 1 1 1 2 0 1 3 1 2 6 2 3 10 8 6 28 18 14 63 61 34 190 170 95 546 538 280 1708 1654 854 5346 5344 2694 17428 17252 8714 m=4 Kn 1 3 5 7 9 11 13 15 17 19 21 23 25 He n 1 1 1 1 4 1 1 1 7 4 0 1 10 10 6 3 13 44 28 11 16 197 193 52 19 1228 1140 307 22 7692 7688 1936 25 52828 52364 13207 28 373636 373560 93496 31 2735952 2732836 683988 34 20506258 20506254 5127163 37 156922676 156899748 39230669 n K en m=6 Kn H Kn Hen 1 1 1 1 1 1 3 0 1 6 3 2 19 10 7 57 54 19 258 222 86 1110 1107 372 5475 5346 1825 27429 27399 9143 143379 142770 47801 764970 764967 254990 4173906 4170672 1391302 Ke n n m=5 Kn Hen 1 1 1 1 5 1 1 1 9 5 0 1 13 15 10 3 17 85 60 17 21 510 505 102 25 4051 3876 811 29 33130 33125 6626 33 291925 290700 58385 37 2661255 2661100 532251 41 25059670 25049020 5011934 45 241724380 241724375 48344880 49 2379912355 2379812100 475982471 n en K m=3 en K en m=7 Kn He n 1 1 1 1 1 1 1 1 6 1 1 1 7 1 1 1 11 6 0 1 13 7 0 1 16 21 15 4 19 28 21 4 21 146 110 25 25 231 182 33 26 1101 1095 187 31 2100 2093 300 31 10632 10326 1772 37 23884 23394 3412 36 107062 107056 17880 43 285390 285383 40770 41 1151802 1149126 191967 49 3626295 3621150 518043 46 12845442 12845166 2141232 55 47813815 47813367 6830545 51 147845706 147817170 24640989 61 650367788 650302814 92909684 Table 1: Number of m-ary and m-gonal cacti having p polygons and n vertices. 23 5 Related enumerative results 5.1 Plane m-gonal cacti Let H denote the one-sort species of plane m-gonal cacti (not m-colored). The case of an isolated vertex is included. If H and A denote the species of pointed and of planted plane m-gonal cacti, respectively, then A coincides with the species introduced in section 2:3, characterized by the functional equation A = XL(Am?1), and H is isomorphic to the species K , for any i, that is, satis es H = X (1 + C (Am?1)): (83) See (26) et (28). However the species H3 of rooted (at a polygon) plane m-gonal cacti is no longer asymmetric. In fact, we have H3 = Cm(A); (84) where Cm denotes the species of circular permutations of length m. Another important di erence resides in the form of the dissymetry theorem which is more closely related to that of free (non plane) m-gonal cacti. Indeed, we have (see [10] and [1], (4.2.16) and Figure 4.2.5) H + H3 = H + A
Am?1 (85) from which we deduce, since Am = A ? X , that H = H + H3 ? A + X = X (1 + C (Am?1 )) + Cm (A) ? A + X: (86) Theorem 26 Let p be a positive integer and set n = p(m ? 1) + 1. Then the numbers Hn and Hen of labelled and unlabelled m-gonal cacti, repectively, having n vertices (and p polygons) are given by ! ( n ? 1)! mp Hn = mp p ; (87) i and where Hen = 1 X ( p )  n = He n = mp djp d ! dm ; d ! pm=d (d) (p ? 1)=d ; dj(m;p?1) ! 1 mp n = Aen = n p : e 3n = 1 n=H mp and n + n ? n; X (88) (89) (90) (91) 2 In the case where m = 2, we recover formulas of Walkup [23] for the number of plane trees. See also Labelle and Leroux ([16], (1.18){(1.21)). It is also possible to derive similar formulas for the number of m-gonal plane cacti according to the vertex-degree distribution. See [16], (1.23){(1.26) where the computations have been carried out in the case m = 2. Table 1 contains numerical values of He n , for n = (m ? 1)p + 1, and m = 2; : : :; 7. 24 ~n (7; 7) (5; 6) (6; 6; 7) (4; 4; 5) (5; 6; 8) (5; 5; 5) (4; 6; 7) (5; 6; 6) (3; 4; 4; 5) (6; 6; 6; 7) Ke~n 226512 5292 28224 225 10584 1323 1960 5488 50 21952 Ke~n 17424 536 3138 39 1176 189 248 692 10 2752 K~n 17424 523 3135 36 1176 189 242 680 10 2736 ~n (1; 3; 3) (2; 2; 3) (1; 4; 4) (2; 3; 4) (3; 3; 3) (3; 3; 5) (1; 3; 3; 3) (2; 2; 3; 3) (2; 3; 4; 4) (4; 4; 4; 4) Ke ~n Ke ~n K~n 1 1 0 3 1 1 1 1 0 6 2 1 16 4 4 20 4 4 1 1 0 3 1 1 6 2 1 125 25 25 Table 2: The number of unlabelled m-ary cacti (rooted, plain, asymmetric) according to their vertex-color distribution. m 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 N Ke N ; i = 1;


; m (1532 ; 27) (8, 7) 2 (1 2241 ; 1224 ) (76, 90) (1323 ; 1323; 1631) (600, 600, 702) (1221 ; 1221; 1221) (12, 12, 12) 4 4 (4; 1 ; 1 ) (1,1,1) (22; 122; 14) (1,2,2) (11 31; 122; 14) (2,3,4) (1222 ; 1222; 1421) (54, 54, 69) (1321 41; 1323; 1721) (600, 720, 960) (1322 ; 1322; 1322) (280, 280, 280) (1232 ; 1422; 1621) (120, 180, 212) 4 4 2 6 1 (2 ; 1 2 ; 1 2 ) (20, 30, 36) (1441 ; 1422; 1422) (252, 300, 300) (1223 ; 1422; 1422) (504, 600, 600) 4 (1 22; 1422; 1422 ; 1621 ) (6000, 6000, 6000, 7008) i Ke N3 14 150 900 16 1 2 4 81 1080 392 240 40 400 800 8000 Ke N KN 1 16 102 4 1 1 1 15 120 56 32 6 52 104 1008 1 14 99 4 0 0 1 12 120 56 28 4 48 96 992 Table 3: The number of unlablelled m-ary cacti (rooted, plain, asymmetric) according to their vertex-degree distributions. 5.2 Free (labelled) m-ary cacti A free m-ary cactus can be informally de ned as an m-ary cactus without the plane embedding. In other words, the m-gons attached to a vertex are free to take any position with respect to each other. Denoting by F the species of free m-ary cacti, we have the functional equations and, for i = 1; 2;


; m, F 3 = A1A2


Am ; (92) Ai = XiE (Abi); (93) 25 where E denotes the species of sets, for which E (x) = ex ; Ee(x) = (1 ? x)?1 and ZE (x1 ; x2;


) = exp ( X xi ): i1 i (94) The computations of subsection 4.1 for labelled m-ary cacti according to vertex-color distribution can be easily adapted to free m-ary cacti. In particular, we nd the following result. Proposition 27 Let ~n = (n1 ;


; nm) be a vector of positive integers satisfying the coherence conditions of Lemma 8 with p = (n ? 1)=(m ? 1)  1. Then the number F (~n) of labelled free m-ary cacti having vertex-color distribution ~n is given by m (n ? 1)!np?n Y i m ? 2 i : (95) F (~n) = p ( p ? n )! i i=1 i 2 This extends Scoins [19] formula nn1 2 ?1 nn2 1 ?1 for the number of labelled bicolored free trees with vertex-color distribution (n1 ; n2) to general m  2. 5.3 Constellations 1 3 1 2 2 1 1 2 3 3 1 3 1 1 2 2 2 3 1 3 3 Figure 9: A rooted ternary constellation. Another combinatorial object closely related to m-ary cacti is an m-ary constellation which is de ned in a similar way as an m-ary cactus except that cycles of polygons are now allowed. Figure 9 shows a typical ternary constellation which is rooted, that is, has a distinguished polygon. M. Bousquet-Melou and G. Schae er [4] have found that the number Ce 3 (p) of unlabelled rooted m-ary constellations having p polygons is given by ! p?1 ( m + 1) m mp 3 Ce (p) = ((m ? 1)p + 2)((m ? 1)p + 1) p : (96) 26 Added in proof For the vertex color distribution ~n = (m; m; : : :; m), numerical computations show that the number Ke ~n of unlabelled m-ary cacti is (m + 1)m?2 . The only vertex degree distribution in this case is (1m?1 2; 1m?12; : : :; 1m?1 2). In terms of topological classi cation of polynomials, this corresponds to the generic case of a polynomial of degree m + 1 whose derivative has m distinct roots. The formula is easily established since there is an obvious bijection between these m-ary cacti and (free) trees with m labelled edges, whose number is (m + 1)m?2; indeed, pointing an arbitrary vertex leads to a tree with m + 1 labelled vertices and Cayley's formula (m +1)m?1 can be invoked. This fact is also mentioned in V.I. 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