arXiv subjects

Wenjie Fang

Publications and source records attributed to Wenjie Fang.

35 records · Page 2Linked to original sources

arXiv2019

A partial order on Motzkin paths

The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a possibility by defining and studying an analogy of the Tamari lattice on Motzkin paths. While our generalization is not a lattice, each of its connected components is isomorphic to an interval in the classical Tamari lattice. With this structural result, we proceed to the enumeration of components and intervals in the poset of Motzkin paths we defined. We also extend the structural and enumerative results to Schröder paths. We conclude by a discussion on the relation between our work and that of Baril and Pallo (2014).

math.CO↗

arXiv2019

The Steep-Bounce Zeta Map in Parabolic Cataland

As a classical object, the Tamari lattice has many generalizations, including $ν$-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to $ν$-Tamari lattices for bounce paths $ν$. We then introduce a new combinatorial object called `left-aligned colorable tree', and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the Steep-Bounce Conjecture using a generalization of the famous zeta map in $q,t$-Catalan combinatorics. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection.

math.CO↗

arXiv2019

Absolute root separation

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic.

math.CA↗

arXiv2019

Fighting fish and two-stack sortable permutations

In 2017, Duchi, Guerrini, Rinaldi and Schaeffer proposed a new family of combinatorial objects called "fighting fish", which are counted by the same formula as more classical objects, such as two-stack sortable permutations and non-separable planar maps. In this article, we explore the bijective aspect of fighting fish by establishing a bijection to two-stack sortable permutations, using a new recursive decomposition of these permutations. With our bijection, we give combinatorial explanations of several results on fighting fish proved previously with generating functions. Using the decomposition of two-stack sortable permutations, we also prove the algebraicity of their generating function, extending a result of Bousquet-Mélou (1998).

math.CO↗

arXiv2018

Subcritical random hypergraphs, high-order components, and hypertrees

In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from $(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest component increases smoothly from $O(\varepsilon^{-2}\log(\varepsilon^3 n))$ to $Θ(n^{2/3})$ and then to $(1 \pm o(1)) 2 \varepsilon n$. As a natural generalisation of random graphs and connectedness, we consider the binomial random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$ (where each $k$-tuple of vertices is present as a hyperedge with probability $p$ independently) and the following notion of high-order connectedness. Given an integer $1 \leq j \leq k-1$, two sets of $j$ vertices are called \emph{$j$-connected} if there is a walk of hyperedges between them such that any two consecutive hyperedges intersect in at least $j$ vertices. A $j$-connected component is a maximal collection of pairwise $j$-connected $j$-tuples of vertices. Recently, the threshold for the appearance of the giant $j$-connected component in $\mathcal{H}^k(n,p)$ and its order were determined. In this article, we take a closer look at the subcritical random hypergraph. We determine the structure, order, and size of the largest $j$-connected components, with the help of a certain class of `hypertrees' and related objects. In our proofs, we combine various probabilistic and enumerative techniques, such as generating functions and couplings with branching processes. Our study will pave the way to establishing a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical random hypergraph by deleting its giant $j$-connected component.

math.CO↗

arXiv2018

Planar triangulations, bridgeless planar maps and Tamari intervals

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 serve as a nexus of bijective links among all these equi-enumerated objects.

math.CO↗

arXiv2017

A trinity of duality: non-separable planar maps, $β$-(1,0) trees and synchronized intervals

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 is closed by map duality. In this article, we show that we can identify the notions of duality in these three classes of objects using previously known natural bijections, which leads to a bijective proof of a result from Kitaev and de Mier (2013). We also discuss how various statistics are transferred by duality and by the bijections we used.

math.CO↗

arXiv2016

The enumeration of generalized Tamari intervals

Let $v$ be a grid path made of north and east steps. The lattice $\rm{T{\scriptsize AM}}(v)$, based on all grid paths weakly above $v$ and sharing the same endpoints as $v$, was introduced by Préville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case $v=(NE)^n$. Our main contribution is that the enumeration of intervals in $\rm{T{\scriptsize AM}}(v)$, over all $v$ of length $n$, is given by $\frac{2 (3n+3)!}{(n+2)! (2n+3)!}$. This formula was first obtained by Tutte(1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in $\rm{T{\scriptsize AM}}(v)$ to non-separable planar maps.

math.CO↗

arXiv2016

Enumerative and bijective aspects of combinatorial maps: generalization, unification and application (PhD thesis)

This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called constellations provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumeration can also help us count other combinatorial objects. This thesis is a sampling from the rich results and connections in the enumeration of maps. This thesis is structured into four major parts. The first part, including Chapter 1 and 2, consist of an introduction to the enumerative study of maps. The second part, Chapter 3 and 4, contains my work in the enumeration of constellations, which are a special type of maps that can serve as a unifying model of some factorizations of the identity in the symmetric group. The third part, composed by Chapter 5 and 6, shows my research on the enumerative link from maps to other combinatorial objects, such as generalizations of the Tamari lattice and random graphs embeddable onto surfaces. The last part is the closing chapter, in which the thesis concludes with some perspectives and future directions in the enumerative study of maps.

math.CO↗

arXiv2016

Cubic graphs and related triangulations on orientable surfaces

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5(g-1)/2-1}γ^{2n}(2n)!$, where $γ$ is an algebraic constant and $c_g$ is a constant depending only on the genus $g$. We also derive an analogous result for simple cubic graphs and weighted cubic multigraphs. Additionally we prove that a typical cubic multigraph embeddable on $\mathbb{S}_g$, $g\ge 1$, has exactly one non-planar component.

math.CO↗

arXiv2015

A recursive structure of sand pile model and its applications

The Sand Pile Model (SPM) and its generalization, the Ice Pile Model (IPM), originate from physics and have various applications in the description of the evolution of granular systems. In this article, we deal with the enumeration and the exhaustive generation of the accessible configuration of the system. Our work is based on a new recursive decomposition theorem for SPM configurations using the notion of staircase bases. Based on this theorem, we provide a recursive formula for the enumeration of SPM(n) and a constant amortized time (CAT) algorithm for the generation of all SPM(n) configurations. The extension of the same approach to the Ice Pile Model is also discussed.

math.CO↗

arXiv2015

Generating functions of bipartite maps on orientable surfaces

We compute, for each genus $g\geq 0$, the generating function $L_g\equiv L_g(t;p_1,p_2,\dots)$ of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L_g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $F_g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet/Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result complements recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d'enfants. Our proofs borrow some ideas from Eynard's "topological recursion" that he applied in particular to even-faced maps (unconventionally called "bipartite maps" in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.

math.CO↗

arXiv2014

Bijective proofs of character evaluations using trace forest of the jeu de taquin

Irreducible characters in the symmetric group are of special interest in combinatorics. They can be expressed either combinatorially with ribbon tableaux, or algebraically with contents. In this paper, these two expressions are related in a combinatorial way. We first introduce a fine structure in the famous jeu de taquin called "trace forest", with which we are able to count certain types of ribbon tableaux, leading to a simple bijective proof of a character evaluation formula in terms of contents that dates back to Frobenius (1901). Inspired by this proof, we give an inductive scheme that gives combinatorial proofs to more complicated formulae for characters in terms of contents.

math.CO↗

arXiv2013

A generalization of the quadrangulation relation to constellations and hypermaps

Constellations and hypermaps generalize combinatorial maps, i.e. embedding of graphs in a surface, in terms of factorization of permutations. In this paper, we extend a result of Jackson and Visentin (1990) stating an enumerative relation between quadrangulations and bipartite quadrangulations. We show a similar relation between hypermaps and constellations by using a result of Littlewood on factorization of characters. A combinatorial proof of Littlewood's result is also given. Furthermore, we show that coefficients in our relation are all positive integers, hinting possibility of a combinatorial interpretation. Using this enumerative relation, we recover a result on the asymptotic behavior of hypermaps in Chapuy (2009).

math.CO↗

arXiv2013

On the Hyperbolicity of Small-World and Tree-Like Random Graphs

Hyperbolicity is a property of a graph that may be viewed as being a "soft" version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. We consider Gromov's notion of δ-hyperbolicity, and establish several results for small-world and tree-like random graph models. First, we study the hyperbolicity of Kleinberg small-world random graphs and show that the hyperbolicity of these random graphs is not significantly improved comparing to graph diameter even when it greatly improves decentralized navigation. Next we study a class of tree-like graphs called ringed trees that have constant hyperbolicity. We show that adding random links among the leaves similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides one of the first significant analytical results on the hyperbolicity of a rich class of random graphs, which shed light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.

cs.SI↗

arXiv2012

New Computational Result on Harmonious Trees

Graham and Sloane proposed in 1980 a conjecture stating that every tree has a harmonious labelling, a graph labelling closely related to additive base. Very limited results on this conjecture are known. In this paper, we proposed a computational approach to this conjecture by checking trees with limited size. With a hybrid algorithm, we are able to show that every tree with at most 31 nodes is harmonious, extending the best previous result in this direction.

cs.DM↗

arXiv2010

A Computational Approach to the Graceful Tree Conjecture

Graceful tree conjecture is a well-known open problem in graph theory. Here we present a computational approach to this conjecture. An algorithm for finding graceful labelling for trees is proposed. With this algorithm, we show that every tree with at most 35 vertices allows a graceful labelling, hence we verify that the graceful tree conjecture is correct for trees with at most 35 vertices.

cs.DM↗