Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1, Q_{n+1}=[x+nz+(y+t)(n+y\partial_y)]Q_n. In this paper we prove Chapoton's conjecture on the duality formula: Q_n(x,y,z,t)=Q_n(x+nz+nt,y,-t,-z), and answer his question about the combinatorial interpretation of Q_n. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating trees and plane trees.
In this paper, we expand on the work of Guo and Zeng from 2007 on a generalization of the Ramanujan polynomials and planar trees. We manage to find combinatorial interpretations of this family of polynomials in terms of Greg trees, Cayley trees, and planar trees by constructing bijections that preserve relevant tree statistics.
By studying non-commutative series in an infinite alphabet we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers-Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers-Ramanujan continued fraction. By specializing the noncommutative series to $q$-series we obtain new combinatorial interpretations to the Rogers-Ramanujan identities in terms of signed integer compositions. We introduce the operation of shift-plethysm on non-commutative ser…
A Grammatical Calculus for the Ramanujan Polynomials
Center for Applied Mathematics, KL-AAGDM Tianjin University（天津大学应用数学中心，KL-AAGDM）；School of Mathematics Shanghai University of Finance and Economics（上海财经大学数学学院）
We use the representation theory of the quasisplit form G of SU(3) over a p-adic field to investigate whether certain quotients of the Bruhat--Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with G (which is a biregular bigraph) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of PGL(2) considered by Lubotzky-Phillips-Sarnak. As a consequence, the classification by Rogawski of the automorphic spectrum of U(3) implies the existence of certain infinite families of Ramanujan b…
Ramanujan defined the polynomials $ψ_{k}(r,x)$ in his study of power series inversion. Berndt, Evans and Wilson obtained a recurrence relation for $ψ_{k}(r,x)$. In a different context, Shor introduced the polynomials $Q(i,j,k)$ related to improper edges of a rooted tree, leading to a refinement of Cayley's formula. He also proved a recurrence relation and raised the question of finding a combinatorial proof. Zeng realized that the polynomials of Ramanujan coincide with the polynomials of Shor, and that the recurrence relation of Shor coincides with the recurrence relation of Berndt, Evans and …
The Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. In an approach to the Cayley formula on the number of trees, Shor discovers a refined recurrence relation in terms of the number of improper edges, without realizing the connection to the Ramanujan polynomials. On the other hand, Dumont and Ramamonjisoa independently take the grammatical approach to a sequence associated with the Ramanujan polynomials and have reached the same conclusion as Shor's. It was a coincidence for Zeng to realize that the Shor polynomials turn out to be the Ramanujan polyno…
A. Lacasse conjectured a combinatorial identity in his study of learning theory. Various people found independent proofs. Here is another one that is based on the study of the tree function, with links to Lamberts $W$-function and Ramanujan's $Q$-function. It is particularly short.
Let $G$ be a large-girth $d$-regular graph and $μ$ be a random process on the vertices of $G$ produced by a randomized local algorithm. We prove the upper bound $(k+1-2k/d)\Bigl(\frac{1}{\sqrt{d-1}}\Bigr)^k$ for the (absolute value of the) correlation of values on pairs of vertices of distance $k$ and show that this bound is optimal. The same results hold automatically for factor of i.i.d processes on the $d$-regular tree. In that case we give an explicit description for the (closure) of all possible correlation sequences. Our proof is based on the fact that the Bernoulli graphing of the infin…
Let $X$ be an infinite graph of bounded degree; e.g., the Cayley graph of a free product of finite groups. If $G$ is a finite graph covered by $X$, it is said to be $X$-Ramanujan if its second-largest eigenvalue $λ_2(G)$ is at most the spectral radius $ρ(X)$ of $X$, and more generally $k$-quasi-$X$-Ramanujan if $λ_k(G)$ is at most $ρ(X)$. In case $X$ is the infinite $Δ$-regular tree, this reduces to the well known notion of a finite $Δ$-regular graph being Ramanujan. Inspired by the Interlacing Polynomials method of Marcus, Spielman, and Srivastava, we show the existence of infinitely many $k$…
We study some sequences of polynomials that appear when we consider the successive derivatives of the tree function (or Lambert's W function). We show in particular that they are related with a generalization of Cayley trees, called Greg trees. Besides the combinatorial result in itself, it is interesting to see how this is related with previous work: similar problems were considered first by Ramanujan, and more recently in the theory of completely monotonic functions and its link with probability. Also of great interest is the fact that these Greg trees were introduced in a problem of textual…
Phylogenetic trees are binary nonplanar trees with labelled leaves, and plane oriented recursive trees are planar trees with an increasing labelling. Both families are enumerated by double factorials. A bijection is constructed, using the respective representations a 2-partitions and trapezoidal words.
We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that if most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. Kesten showed that if a Cayley …
In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges.
A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that …
Center for Applied Mathematics, KL-AAGDM Tianjin University（天津大学应用数学中心）；College of Mathematical Science Institute of Mathematics and Interdisciplinary Sciences Tianjin Normal University（天津师范大学数学科学学院）
Let $T(n,m)$ be the set of all plane labelled bipartite trees with $n$ white vertices and $m$ black. If the number $n+m$ of vertices is even, then the set $T(n,m)$ is a union of two disjoint subsets --- subset od "even" trees and subset of "odd" trees. This partition has a clear geometric meaning.
A type is the set of all pairwise nonisotopic binary trees with the same passport. A type is called decomposable, if it is a union of several Galois orbits. In this work we present the list of all passports of plane binary trees with ten edges and the list of all Galois orbits.
We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size $n$ approaches 0.727649, and the average rank of the root of a tree of size $n$ approaches 1.62297.
The protection number of a plane tree is the minimal distance of the root to a leaf; this definition carries over to an arbitrary node in a plane tree by considering the maximal subtree having this node as a root. We study the the protection number of a uniformly chosen random tree of size $n$ and also the protection number of a uniformly chosen node in a uniformly chosen random tree of size $n$. The method is to apply singularity analysis to appropriate generating functions. Additional results are provided as well.
We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.
Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.
We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem on free Dyck paths and some genera…
Some of Philippe Flajolet's combinatorial contributions that he wrote between 1976 and 1995, say, are described. In most of Flajolet's papers, asymptotic/analytic considerations play a major role. To be true to the spirit of the journal ECA, the emphasis is on the \emph{combinatorial} part. Covered are Register function of binary trees, approximate counting, digital search trees and the Rice formula, probabilistic counting, adding a new slice, Ramanujan's influence, finite differences and harmonic numbers, Digits and the Mellin-Perron formula.
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with $n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.
We count the number of vertices in plane trees and $k$-ary trees with given outdegree, and prove that the total number of vertices of outdegree $i$ over all plane trees with $n$ edges is ${2n-i-1 \choose n-1}$, and the total number of vertices of outdegree $i$ over all $k$-ary trees with $n$ edges is ${k\choose i}{kn\choose n-i}$. For both results we give bijective proofs as well as generating function proofs.
A 2-tree is a graph that can be formed by starting with a triangle and iterating the operation of making a new vertex adjacent to two adjacent vertices of the existing graph. Leizhen Cai asked in 1995 whether every maximal planar graph contains a spanning 2-tree. We answer this question in the negative by constructing an infinite class of maximal planar graphs that have no spanning 2-tree.
Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems.
In response to a well-known open question ``Does every complete geometric graph on $2n\/$ vertices have a partition of its edge set into $n\/$ plane spanning trees?" we provide an affirmative answer when the complete geometry graph is in the regular wheel configuration. Also we present sufficient conditions for the complete geometric graph on $2n\/$ vertices to have a partition of its edge set into $n\/$ plane spanning trees (which are double stars, caterpillars or $ w\/$-caterpillars).
The problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and Simion in their study of Schröder paths, and it was recently derived by Coker using the Lagrange inversion formula. An equivalent problem for partitions was independently studied by Klazar. We present three parity reversing involutions, one for unlabelled plane trees, the other for labelled plane trees and one for 2-Motzkin paths which are in one-to-one corr…
Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is C_{k,m}(n)=\frac{1}{mn+1}{(mn+1)k \choose n}, which we call the (k,m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k,m)-ary trees, which are counted by the (k,m)-Catalan numbers.
In 2005 Janson, extending earlier work of Mahmoud, Smythe, and Szymański, established the joint asymptotic normality of the outdegrees of a random plane recursive tree. In particular, he gave an explicit description of the limiting covariance matrix. Our aim here is to provide substantially simplified expression for the limiting covariance matrix.
Nesetril and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On the positive side, every first order convergent sequence of trees or graphs with no long path (graphs with bounded tree-depth) has a limit modeling. We strengthen these results by showing that every first order convergent sequence of plane trees (trees with embeddings in the plane) and every first or…
By weighted tree we understand such connected tree,that: a) each its vertex and each edge have a positive integer weight; b) the weight of each vertex is equal to the sum of weights of outgoing edges. Each tree has a binary structure --- we can color its vertices in two colors, black and white so, that adjacent vertices have different colors. A type is a set of pairwise non-isotopic plane weighted trees with a given list of weights of white vertices and a given list of weights of black vertices. In this work we present a method for computing the cardinality of a given type.
A leaf of a plane tree is called an old leaf if it is the leftmost child of its parent, and it is called a young leaf otherwise. In this paper we enumerate plane trees with a given number of old leaves and young leaves. The formula is obtained combinatorially by presenting two bijections between plane trees and 2-Motzkin paths which map young leaves to red horizontal steps, and old leaves to up steps plus one. We derive some implications to the enumeration of restricted permutations with respect to certain statistics such as pairs of consecutive deficiencies, double descents, and ascending run…
We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0, whose combinatorial properties are closely connected with empirical distributions, plane trees, plane partitions, parking functions, and the associahedron. In particular, we give explicit formulas for the volume of Pi_n(x) and, when the x_i's are integers, the number of integer points in Pi_n(x). We give two polyhedral decompositions of Pi_n(x), one related to order cones of posets and the other to the associahedron.
We present new functional equations for the species of plane and of planar (in the sense of Harary and Palmer, 1973) 2-trees and some associated pointed species. We then deduce the explicit molecular expansion of these species, i.e a classification of their structures according to their stabilizers. There result explicit formulas in terms of Catalan numbers for their associated generating series, including the asymmetry index series. This work is closely related to the enumeration of polyene hydrocarbons of molecular formula C_nH_n+2.
We define the disposition polynomial $R_{m}(x_1, x_2, ..., x_n)$ as $\prod_{k=0}^{m-1}(x_1+x_2+...+x_n+k)$. When $m=n-1$, this polynomial becomes the generating function of plane trees with respect to certain statistics as given by Guo and Zeng. When $x_i=1$ for $1\leq i\leq n$, $R_{m}(x_1, x_2, ..., x_n)$ reduces to the rising factorial $n(n+1)... (n+m-1)$. Guo and Zeng asked the question of finding a combinatorial proof of the formula for the generating function of plane trees with respect to the number of younger children and the number of elder children. We find a combinatorial interpretat…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on $\{1,2,\ldots,n\}$. Classical formulas for counting various trees such as rooted trees, bipartite trees, tripartite trees, plane trees, $k$-ary plane trees, $k$-edge colored trees follow immediately from our recursive relations.
The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge rotations). In a transition graph on the set $\mathcal{T}(S)$ of noncrossing straight-line spanning trees on a finite point set $S$ in the plane, two spanning trees are connected by an edge if one can be transformed into the other by such an operation. We study bounds on the diameter of these graphs, and consider the various operations on point sets in both g…
Recently Han obtained a general formula for the weight function corresponding to the expansion of a generating function in terms of hook lengths of binary trees. In this paper, we present formulas for k-ary trees, plane trees, plane forests, labeled trees and forests. We also find appropriate generating functions which lead to unifications of the hook length formulas due to Du and Liu, Han, Gessel and Seo, and Postnikov.
We prove that any planar graph on $n$ vertices has less than $O(5{.}2852^n)$ spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less than $O(2{.}7156^n)$. As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by $O(147.{7}^n)$. Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by $O(6.4884^n)$, the number of plane spanning trees on a set of $n$ poin…
Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., th…
We define a collection of functions $s_i$ on the set of plane trees (or standard Young tableaux). The functions are adapted from transpositions in the representation theory of the symmetric group and almost form a group action. They were motivated by $\textit{local moves}$ in combinatorial biology, which are maps that represent a certain unfolding and refolding of RNA strands. One main result of this study identifies a subset of local moves that we call $s_i$-local moves, and proves that $s_i$-local moves correspond to the maps $s_i$ acting on standard Young tableaux. We also prove that the gr…
In \cite{BaDeFePi96} the concept of nondecreasing Dyck paths was introduced. We continue this research by looking at it from the point of view of words, rational languages, planted plane trees, and continued fractions. We construct a bijection with planted plane trees of height $\le 4$ and compute various statistics on trees that are the equivalents of nondecreasing Dyck paths.
We prove that for any fixed $k$, the probability that a random vertex of a random increasing plane tree is of rank $k$, that is, the probability that a random vertex is at distance $k$ from the leaves, converges to a constant $c_k$ as the size $n$ of the tree goes to infinity. {\color{blue} We prove that $1-\sum_{j\le k} c_k<\tfrac{3^{k+1}}{(2k+1)!}$, so that the tail of the limiting rank distribution is super-exponentially narrow. We prove that the latter property holds uniformly for all finite $n$ as well.} More generally, we prove that the ranks of a finite uniformly random set of vertices …
