The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, projective plane and the Klein bottle. We also present the results of enumerating some special kinds of maps on the sphere: near-$r$-regular maps, maps with multiple leaves and maps with multiple root semi-edges. For $r=3$ and $r=4$ we obtain exact analytical formulas. For larger $r$ we derive recurrence relations. Then using these results we enumerate $r$-regular maps on the torus up to homeomorphisms that preserve its orientation --- so-called sensed maps. Using the concept of a quotient map on an orbifold we reduce this problem to enumeration of certain classes of rooted maps. For $r=3$ and $r=4$ we obtain closed-form expressions for the numbers of $r$-regular sensed maps by edges. All these results will be used in the second part of the work to enumerate $r$-regular maps on the torus up to all homeomorphisms --- so-called unsensed maps.
We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Faà di Bruno formula, and model the generation of ballot and Dyck paths. We get formulas for its enumeration according with the height of their lattice points. Recursive formulas for the enumeration of enriched increasing trees and forests with respect to the height of their internal and external vertices are also obtained. Finally we present a generalized form of all those results …
In the work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. In this work a marked directed edge is not demanded, i.e. we enumerate tree-rooted planar cubic maps. Formulas are more complex, of course, but not significantly.
Let $\T_{n}$ be the set of rooted labeled trees on $\set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new refinement $\T_{n,k}$ of $\T_n$, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices.
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $\ge k$ at level $\ge \ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.
We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted trees on the vertex set $[n+1]$ in which exactly $k$ children of the root are lower-numbered than the root is $\binom{n}{k} \, n^{n-k}$. Here I give a simpler proof of this result.
In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree $k$ at level $\ell$, (ii) non-first-children of degree $k$ at level $\ell-1$, (iii) leaves having $k-1$ elder siblings at level $\ell$, and (iv) non-leaves of outdegree $k$ at level $\ell-1$. Our results unite and generalize several previous works in the literature.
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.
This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root dart are distinguished. We give functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the vertices containing the distinguished darts. We solve these equations to get parametric expressions of the generating functions of rooted hypermaps of low genus. We a…
A one-to-one correspondence is proved between the N-rooted ribbon graphs, or maps, with e edges and the (e-N+1)-loop Feynman diagrams of a certain quantum field theory. This result is used to obtain explicit expressions and relations for the generating functions of N-rooted maps and for the numbers of N-rooted maps with a given number of edges using the path integral approach applied to the corresponding quantum field theory.
Fano Bott manifolds bijectively correspond to signed rooted forests with some equivalence relation. Using this bijective correspondence, we enumerate the isomorphism classes of Fano Bott manifolds and the diffeomorphism classes of indecomposable Fano Bott manifolds. We also observe that the signed rooted forests with the equivalence relation bijectively correspond to rooted triangular cacti.
This paper discusses the enumeration for the total number of all rooted spanning forests of the labeled complete tripartite graph. We enumerate the total number by a combinatorial decomposition.
$β(1,0)$-trees provide a convenient description of rooted non-separable planar maps. The involution $h$ on $β(1,0)$-trees was introduced to prove a complicated equidistribution result on a class of pattern-avoiding permutations. In this paper, we describe and enumerate fixed points of the involution $h$. Intriguingly, the fixed points are equinumerous with the fixed points under taking the dual map on rooted non-separable planar maps, even though the fixed points do not go to each other under the know (natural) bijection between the trees and the maps.
We enumerate rooted triangulations of a sphere with multiple holes by the total number of edges and the length of each boundary component. The proof relies on a combinatorial identity due to W.T. Tutte.
In the present work the rooted and unrooted d-regular maps on 2-dimentional oriented surfaces of genus g are enumerated. Separately and in more detail the case of d-regular maps with a single face are considered.
Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula to count the number of distinct binary rooted tanglegrams with $n$ matched vertices, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This includes a new formula for the number of binary trees with…
A recursive method is given for finding generating functions which enumerate rooted hypermaps by number of vertices, edges and faces for any given number of darts. It makes use of matrix-integral expressions arising from the study of bipartite quantum systems. Direct evaluation of these generating functions is then demonstrated through the enumeration of all rooted hypermaps with up to 13 darts.
We enumerate rooted 2-connected and 3-connected surface maps with respect to vertices and edges. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled graphs of genus g (1-connected, 2-connected, and 3-connected graphs.)
We present several bijections, in terms of combinatorial objects counted by the Schröder numbers, that are then used (via coloring) for the construction and enumeration of rational Schröder paths with integer slope, ordered rooted trees, and simple rooted outerplanar maps. On the other hand, we derive partial Bell polynomial identities for the little and large Schröder numbers, which allow us to obtain explicit enumeration formulas.
In this work for the first time we enumerate unlabelled maps on orientable genus $g$ surfaces with respect to all homeomorphisms, including both orientation-preserving and orientation-reversing. We show that in the latter case as an intermediate step one has to enumerate rooted maps of a special kind (quotient maps) on orientable and non-orientable surfaces possibly having a boundary and a certain number of branch points. In this work we develop a special technique for enumerating such maps.
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.
For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence $λ= 1^{e_1}2^{e_2} \ldots$ of a tree on the vertex set $\set{1,2,\ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corr…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that such mappings achieve much more than enumeration of rooted trees. We discuss two related structural bijections. The first corresponds to a bijective map between integers and rooted trees. The first bijection also suggests a new algorithm for sifting primes. The second bijection extends the first one in order to map rational numbers to a family of rooted trees…
We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs rely on the generalisation to orientable surfaces of the Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We show that each of the 2g fondamental cycles of the surface contributes a factor m between the numbers of m-hypermaps…
Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function $H(x)$ of all 1342-avoiding permutations of length $n$ as well as an {\em exact} formula for their number $S_n(1342)$. While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length $n$ equals that of labeled plane trees of a certain type on $n$ vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, $H(…
The enumeration of maps and the study of uniform random maps have been classical topics of combinatorics and statistical physics ever since the seminal work of Tutte in the sixties. Following the bijective approach initiated by Cori and Vauquelin in the eighties, we describe a bijection between rooted maps, or rooted bipartite quadrangulations, on a surface of genus g and some simpler objects that generalize plane trees. Thanks to a rerooting argument, our bijection allows to compute the generating series of rooted maps on a surface of genus g with respect to the number of edges, and to recove…
In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite circle graphs. Then, we focus on Depth-First Search trees and their connection with a poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the f…
We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent to the $q$-state Potts model randomized over such maps. Like the regular ferromagnetic lattice models, it has a first-order transition when $q$ is greater than a critical value $q_c$, but $q_c$ is much larger - about 72 instead of 4.
We determine the number of labelled chordal planar graphs with $n$ vertices, which is asymptotically $c_1\cdot n^{-5/2} γ^n n!$ for a constant $c_1>0$ and $γ\approx 11.89235$. We also determine the number of rooted simple chordal planar maps with $n$ edges, which is asymptotically $c_2 n^{-3/2} δ^n$, where $δ= 1/σ\approx 6.40375$, and $σ$ is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively r…
Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimss…
