Fred Curtis
Existing planar map encodings neglect maps with loops. The presented scheme encodes any connected planar map in 4 bits/edge. Encoding and decoding time is O(edges). Implicit face/edge/vertex orderings and canonical encodings are discussed.
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We give new equations which characterize the generating functions of planar quadrangulations and planar triangulations, with zero, one or two boundaries. The proof is inspired by the Lackner--Panholzer last car decomposition of parking trees (arXiv:1504.04972) and consists in applying a similar decomposition to the peeling trees of planar maps.
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.
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.
There doesn't exists a finite planar map with all edges having the same length, and each vertex on exactly 5 edges.
We present a direct bijection between planar 3-connected triangulations and bridgeless planar maps, which were first enumerated by Tutte (1962) and Walsh and Lehman (1975) respectively. Previously known bijections by Wormald (1980) and Fusy (2010) are all defined recursively. Our direct bijection passes by a new class of combinatorial objects called "sticky trees". We also present bijections between sticky trees, intervals in the Tamari lattices and closed flows on forests. With our bijections, we recover several known enumerative results about these objects. We thus show that sticky trees can…
Slit-slide-sew bijections for oriented planar maps
本文针对双极有向平面地图与Schnyder woods，提出基于切缝-滑动-缝合操作及重根轨道研究的增长双射构造方法，为相关计数恒等式提供直接组合证明。
This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific "transversal structures" on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijection between rooted loopless maps and rooted triangulations, based on the observation that loopless maps and triangulations are decomposed in a similar way into components that are re…
We present bijections for the planar cases of two counting formulas on maps that arise from the KP hierarchy (Goulden-Jackson and Carrell-Chapuy formulas), relying on a "cut-and-slide" operation. This is the first time a bijective proof is given for quadratic map-counting formulas derived from the KP hierarchy. Up to now, only the linear one-faced case was known (Harer-Zagier recurrence and Chapuy-Féray-Fusy bijection). As far as we know, this bijection is new and not equivalent to any of the well-known bijections between planar maps and tree-like objects.
Enumerating planar stuffed maps as hypertrees of mobiles
本文提出将双色平面填充地图与超移动体之间的双射，推广了平面地图与移动体的双射，并研究了生成函数的代数与函数方程。
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…
In this work we present an algorithm with which any arbitrary cubic planar map may be constructed through successive edge insertion while simultaneously constructing a set of proper edge labels and Hamiltonian cycles for each configuration. We present a publicly available Python implementation of this algorithm, and discuss both theoretical and numerical support for its validity, with reference to the well-known Four Colour Theorem.
Rooted bicubic planar maps via Dyck paths
本文通过组合方法证明了Tutte将根bicubic平面图分解为3连通组件的定理，并建立了根bicubic平面图与具有特定上坡长度的Dyck路径之间的双射关系，同时给出了通过插入操作生成所有根bicubic平面图的方法。
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers $D$. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.
In 2006, Chapoton defined a class of Tamari intervals called "new intervals" in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called "degree tree". Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy.
The class of fighting fish is a recently introduced model of branching surfaces generalizing parallelogram polyominoes. We can alternatively see them as gluings of cells, walks on the square lattice confined to the quadrant or shuffle of Dyck words. With these different points of view, we introduce a natural extension of fighting fish that we call \emph{generalized fighting fish}. We show that generalized fighting fish are exactly the Mullin codes of rooted planar maps endowed with their unique rightmost depth-first search spanning tree, also known as Lehman-Lenormand code. In particular, this…
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss some possible extension to maps of higher genus.
Slit-Slide-Sew bijections for planar bipartite maps with prescribed degree
研究具有规定度数的平面二分图，利用对偶性和 Schaeffer 双射将 Louf 公式简化为树的方程，通过双射证明相关公式，并将树的构造解释为图上的‘slit - slide - sew’操作，首次为源于可积层级的多参数公式给出双射。
Enumeration of rooted 3-connected bipartite planar maps
本研究首次解决有根3-连通二部平面图的计数问题，通过分解图的连通分量得到对应生成函数，推导出该类图数量的渐近估计式。
The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize beta-description trees to cyclicly Gray code three classes of cubic planar maps, namely, bicubic planar maps, 3-connected cubic planar maps, and cubic non-separable planar maps.
We present recent results on the enumeration of $q$-coloured planar maps, where each monochromatic edge carries a weight $ν$. This is equivalent to weighting each map by its Tutte polynomial, or to solving the $q$-state Potts model on random planar maps. The associated generating function, obtained by Olivier Bernardi and the author, is differentially algebraic. That is, it satisfies a (non-linear) differential equation. The starting point of this result is a functional equation written by Tutte in 1971, which translates into enumerative terms a simple recursive description of planar maps. The…
We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. We also obtain a far-reaching generalization for other face-degrees. In fact, all the known enumerative formulas for maps with boundaries are proved bijectively in the present article (and several new formulas are obtained…
This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of α-orientations introduced by Felsner, and in particular on the existence of …
On quasi-polynomials counting planar tight maps
本文利用平面映射的切片分解与双射方法，给出了计数具有指定面度数的平面紧映射的拟多项式显式公式，并将Tutte切片公式推广至所有非二分映射。
These are lecture notes of the 48th Saint-Flour summer school, July 2018, on the topic of planar maps, random walks and the circle packing theorem.
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…
Drmota and Stufler proved recently that the expected number of pattern occurrences of a given map is asymptotically linear when the number of edges goes to infinity. In this paper we improve their result by means of a different method. Our method allows us to develop a systematic way for computing the explicit constant of the linear (main) term and shows that it is a positive rational number. Moreover, by extending our method, we also solve the corresponding problem of submap occurrences.
The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n \to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.
A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no beta-redexes, and planar if it is linear and the use of variables moreover follows a deterministic stack discipline. We begin by showing that the sequence counting normal planar lambda terms by a natural notion of size coincides with the sequence (originally computed by Tutte) counting rooted planar maps by number of edges. Next, we explain how to apply the ma…
Counting colored planar maps: algebraicity results
本文证明当 q 为特定形式（含 q=2,3）时，适当 q 着色平面地图及 Potts 加权地图的生成函数是代数的，并推广至三角剖分，首次求解了带两个催化变量的非线性方程。
Bijections between planar maps and planar linear normal $λ$-terms with connectivity condition
本文通过构造直接双射证明了 3-连通平面线性正规 λ-项与二部平面地图的计数公式相同，并给出新的平面线性正规 λ-项与平面地图双射，其 2-连通限制导出无环平面地图。
Decomposition of Greedy Tamari Intervals and Bipartite Planar Maps
通过建立贪婪Tamari区间与二部平面地图的同构递归分解，给出了贪婪1-Tamari区间与平面2-星座等计数的组合证明，并提出了更一般的猜想。
We introduce the set of (non-spanning) tree-decorated planar maps, and show that they are in bijection with the Cartesian product between the set of trees and the set of maps with a simple boundary. As a consequence, we count the number of tree decorated triangulations and quadrangulations with a given amount of faces and for a given size of the tree. Finally, we generalise the bijection to study other types of decorated planar maps and obtain explicit counting formulas for them.
We consider planar maps adjusted with a (regular critical) Boltzmann distribution and show that the expected number of pattern occurrences of a given map is asymptotically linear when the number n of edges goes to infinity. The main ingredient for the proof is an extension of a formula by Liskovets (1999).
We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order c log(n) for an explicit constant c. These results provide new information on the structure of random planar graphs.
Counting coloured planar maps: differential equations
本文研究按边数和单色边数计数的 q-着色平面地图，证明其生成函数满足关于边变量的多项式微分方程，并给出显式微分系统，同时将结果推广到平面三角剖分，从而在随机平面格点上求解 q-态 Potts 模型。
We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth…
The dual of a map is a fundamental construction on combinatorial maps, but many other combinatorial objects also possess their notion of duality. For instance, the Tamari lattice is isomorphic to its order dual, which induces an involution on the set of so-called "synchronized intervals" introduced by Préville-Ratelle and the present author. Another example is the class of $β$-(1,0) trees, which has a mysterious involution $h$ proposed by Claesson, Kitaev and Steingrímsson (2009). These two classes of combinatorial objects are all in bijection with the class of non-separable planar maps, which…
We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk.
Stephenson~(2018) established annealed local convergence of Boltzmann planar maps conditioned to be large. The present work uses results on rerooted multi-type branching trees to prove a quenched version of this limit.
We compute the distance-dependent three-point function of general planar maps and of bipartite planar maps, i.e., the generating function of these maps with three marked vertices at prescribed pairwise distances. Explicit expressions are given for maps counted by their number of edges only, or by both their numbers of edges and faces. A few limiting cases and applications are discussed.
Enumeration of planar bipartite tight irreducible maps
本文研究平面二部紧且2b-不可约地图的计数问题，通过基于切片分解的直接双射方法，将地图分解为2b-不可约紧切片和双面地图，并利用b-装饰树进行双射编码，最终给出了该类地图数量的显式多项式表达式。
The generating function of planar Eulerian orientations
该研究通过将生成函数表示为简单超几何级数的反函数，解决了平面欧拉定向及4价平面欧拉定向的计数问题，验证了此前的渐近行为预测，并发现了4价情形下的意外组合关联。
We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.
We characterize the generating function of bipartite planar maps counted according to the degree distribution of their black and white vertices. This result is applied to the solution of the hard particle and Ising models on random planar lattices. We thus recover and extend some results previously obtained by means of matrix integrals. Proofs are purely combinatorial and rely on the idea that planar maps are conjugacy classes of trees. In particular, these trees explain why the solutions of the Ising and hard particle models on maps of bounded degree are always algebraic.
Local Limit of Random Regular Bipartite Planar Maps
本文证明了当顶点数趋于无穷时，均匀随机d-正则双部平面图的局部极限存在，通过建立图与开花树的双射关系，并扩展到无限树以转移收敛至平面图。
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…
We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences, to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels.…
The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolation on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations distinct vertices on the exploration contour may at later stage be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative "simple" peeling …
Unified study of the phase transition for block-weighted random planar maps
本文将 Fleurat 与 Salvy 于2024年提出的块加权随机地图相变分析方法推广至更多地图族，证明相变存在并刻画各机制下最大块的尺寸。
We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each $α\in (1,2)$ that if the face degree is in the domain of attraction of an $α$-stable Lévy process, the corresponding random planar map has an infinite volume limit in the Benjamini-Schramm topology. We also show in the limit that the properly rescaled contour functions associated with the northwest and southeast trees converge in law to a certain correlated pair of $α$-stable Lévy processes. Combined with other work, this allows us to identify the scaling limit of the planar map with an SLE$_κ(ρ)$ …
We consider a natural local dynamic on the set of all rooted planar maps with $n$ edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the $n$-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this "edge rotation" chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times $n^{-11/2}$. In doing so, we provide a partially new proof of the fact that …
Characterisation of Markov property on planar maps
该研究从三个视角重新审视平面地图及带装饰平面地图的马尔可夫性质，刻画了满足马尔可夫性质和重根不变性的概率律，全面分析停止地图并引入带装饰的度量平面地图。
Liouville Quantum Duality and Random Planar Maps
本文研究块加权随机平面地图模型，发现其在临界值 $u=u_{cr}$ 处发生退化相变，并证明该临界点与临界点之下的计数性质和临界指数由对偶关系相联系，这与Liouville量子引力描述的预期一致。
We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of the complement of {x} in S. In particular, points …
We consider the problem of computing the distance-dependent two-point function of general planar maps and hypermaps, i.e. the problem of counting such maps with two marked points at a prescribed distance. The maps considered here may have faces of arbitrarily large degree, which requires new bijections to be tackled. We obtain exact expressions for the following cases: general and bipartite maps counted by their number of edges, 3-hypermaps and 3-constellations counted by their number of dark faces, and finally general and bipartite maps counted by both their number of edges and their number o…
We analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest. From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic planar maps, which allow us to recover known results in a more general and transparent way. This approach allows us to obtain new enumerative results. Concerning random maps, we first obtain the distribution of the degree of the root face, which has an exponential tail as for other classes of random maps. Our main result is a limiting map-Airy distribution law fo…
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a…
We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between …
A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of fixed genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer's bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expressio…
On Planarity of Graphs in Homotopy Type Theory
本文在同伦类型论中构造性地发展图论，利用映射、外部面和游走同伦给出有限连通多重图平面性的初等刻画，并在Agda中形式化验证。
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