Non-crossing Trees Research Articles

On individual leaf depths of trees

We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work of Panholzer and Prodinger, we apply this method to study the depth distributions of individual nodes or leaves in rooted binary trees, plane trees, noncrossing trees, and increasing trees; the height distributions of individual vertices and individual steps in Dyck paths; and the number of diagonals separating two fixed sides of a convex polygon in a triangulation or dissection. We obtain both exact and asymptotic results, which sometimes refine known formulas or provide combinatorial proofs of results from the probability literature.

On scaling limits of random Halin-like maps

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the α-stable looptrees of Curien and Kortchemski.

Reachability in complete t-ary trees

Mathematical trees such as Cayley trees, plane trees, binary trees, noncrossing trees, t-ary trees among others have been studied extensively. Reachability of vertices as a statistic has been studied in Cayley trees, plane trees, noncrossing trees and recently in t-ary trees where all edges are oriented from vertices of lower label towards vertices of higher label. In this paper, we obtain closed formulas as well as asymptotic formulas for the number of complete t-ary trees in which there are paths of a given length such that the terminal vertex is a sink, leaf sink, first child and non-first child. We also obtain number of trees in which there is a leftmost path of a given length.

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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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+1$$\\end{document}. These trees are known to be related to (k+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k+1)$$\\end{document}-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 are related to (2k+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2k+1)$$\\end{document}-ary trees.

Bijections of \(k\)-plane trees

A \(k\)-plane tree is a tree drawn in the plane such that the vertices are labeled by integers in the set \(\{1,2,\ldots,k\}\), the children of all vertices are ordered, and if \((i,j)\) is an edge in the tree, where \(i\) and \(j\) are labels of adjacent vertices in the tree, then \(i+j\leq k+1\). In this paper, we construct bijections between these trees and the sets of \(k\)-noncrossing increasing trees, locally oriented \((k-1)\)-noncrossing trees, Dyck paths, and some restricted lattice paths.

On Noncrossing and Plane Tree-Like Structures

Mathematical trees are connected graphs without cycles, loops and multiple edges. Various trees such as Cayley trees, plane trees, binary trees, $d$-ary trees, noncrossing trees among others have been studied extensively. Tree-like structures such as Husimi graphs and cacti are graphs which posses the conditions for trees if, instead of vertices, we consider their blocks. In this paper, we use generating functions and bijections to find formulas for the number of noncrossing Husimi graphs, noncrossing cacti and noncrossing oriented cacti. We extend the work to obtain formulas for the number of bicoloured noncrossing Husimi graphs, bicoloured noncrossing cacti and bicoloured noncrossing oriented cacti. Finally, we enumerate plane Husimi graphs, plane cacti and plane oriented cacti according to number of blocks, block types and leaves.

Non-crossing Trees, Quadrangular Dissections, Ternary Trees, and Duality-Preserving Bijections

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set of labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the set of ternary trees, the set of quadrangular dissections, and the set of flagged Perfectly Chain Decomposed Ditrees, and we give topological and/or combinatorial interpretations of the structural bijections between them. In particular the bijection from the set of quadrangular dissections to the set of non-crossing trees seems to be new. Further we give explicit formulas for the number of self-dual labeled and unlabeled non-crossing trees and the set of quadrangular dissections up to rotations and up to rotations and reflections.

Brussels sprouts, noncrossing trees, and parking functions

Brussels sprouts, noncrossing trees, and parking functions

Oriented Flip Graphs and Noncrossing Tree Partitions

Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.

How to embed noncrossing trees in Universal Dependencies treebanks in a low-complexity regular language

A recently proposed balanced-bracket encoding (Yli-Jyrä and GómezRodríguez 2017) has given us a way to embed all noncrossing dependency graphs into the string space and to formulate their exact arcfactored inference problem (Kuhlmann and Johnsson 2015) as the best string problem in a dynamically constructed and weighted unambiguous context-free grammar. The current work improves the encoding and makes it shallower by omitting redundant brackets from it. The streamlined encoding gives rise to a bounded-depth subset approximation that is represented by a small finite-state automaton. When bounded to 7 levels of balanced brackets, the automaton has 762 states and represents a strict superset of more than 99.9999% of the noncrossing trees available in Universal Dependencies 2.4 (Nivre et al. 2019). In addition, it strictly contains all 15-vertex noncrossing digraphs. When bounded to 4 levels and 90 states, the automaton still captures 99.2% of all noncrossing trees in the reference dataset. The approach is flexible and extensible towards unrestricted graphs, and it suggests tight finite-state bounds for dependency parsing, and for the main existing parsing methods.

Semistable subcategories for tiling algebras

Semistable subcategories were introduced in the context of Mumford’s GIT and interpreted by King in terms of representation theory of finite dimensional algebras. Ingalls and Thomas later showed that for finite dimensional algebras of Dynkin and affine type, the poset of semistable subcategories is isomorphic to the corresponding poset of noncrossing partitions. We show that semistable subcategories defined by tiling algebras, introduced by Coelho Simoes and Parsons, are in bijection with noncrossing tree partitions, introduced by the second author and McConville. Moreover, this bijection defines an isomorphism of the posets on these objects. Our work recovers that of Ingalls and Thomas in Dynkin type A.

ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS

AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

M-Noncrossing trees

We examine [Formula: see text]-cluster theory from an elementary point of view using a generalization of [Formula: see text]-ary trees which we call [Formula: see text]-noncrossing trees. We show that these trees are in bijection with [Formula: see text]-clusters in the [Formula: see text]-cluster category of a quiver of type [Formula: see text]. Similar trees are in bijection with complete exceptional sequences. Most of this paper is expository, explaining definitions and known results about these topics in representation theory. One application of these trees is that the mutation formula for [Formula: see text]-clusters is derived from the more elementary mutation of trees. The main new result is that the natural map of an [Formula: see text]-noncrossing tree into the plane is an embedding. We also explain the relationship between [Formula: see text]-noncrossing trees and finite Harder–Narasimhan systems in the derived category of the module category of type [Formula: see text].

Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions

Given a tree embedded in a disk, we introduce a simplicial complex of noncrossing geodesics supported by the tree, which we call the noncrossing complex. The facets of the noncrossing complex have the structure of an oriented flip graph. Special cases of these oriented flip graphs include the Tamari lattice, type A Cambrian lattices, Stokes posets of quadrangulations, and oriented exchange graphs of quivers mutation-equivalent to a type A Dynkin quiver. We prove that the oriented flip graph is a polygonal, congruence-uniform lattice. To do so, we express the oriented flip graph as a lattice quotient of a lattice of biclosed sets.The facets of the noncrossing complex have an alternate ordering known as the shard intersection order. We prove that this shard intersection order is isomorphic to a lattice of noncrossing tree partitions, which generalizes the classical lattice of noncrossing set partitions. The oriented flip graph inherits a cyclic action from its congruence-uniform lattice structure. On noncrossing tree partitions, this cyclic action generalizes the classical Kreweras complementation on noncrossing set partitions.

Unfoldings of an envelope

An envelope is equivalent to a rectangle dihedron or a doubly-covered rectangle. It is cut along a tree graph that spans the four corners of the envelope to get a planar region. We show in Theorem 1 that every region satisfies the Conway criterion and so copies of the region tile the plane using only translations and 180° rotations. Let P1 and P2 be two regions obtained by unfolding the same envelope along two non-crossing trees, respectively. Then we show in Theorem 2 that P1 is equi-rotational into P2, which means that P1 can be dissected into pieces that are hinged at corners, so that the pieces can be rigidly transformed into P2 by monotonous rotations at the hinges. In Theorems 3 and 4, we give the sufficient conditions for Conway tiles to be foldable into envelopes.

Locally Oriented Noncrossing Trees

We define an orientation on the edges of a noncrossing tree induced by the labels: for a noncrossing tree (i.e., the edges do not cross) with vertices $1,2,\ldots,n$ arranged on a circle in this order, all edges are oriented towards the vertex whose label is higher. The main purpose of this paper is to study the distribution of noncrossing trees with respect to the indegree and outdegree sequence determined by this orientation. In particular, an explicit formula for the number of noncrossing trees with given indegree and outdegree sequence is proved and several corollaries are deduced from it. Sources (vertices of indegree $0$) and sinks (vertices of outdegree $0$) play a special role in this context. In particular, it turns out that noncrossing trees with a given number of sources and sinks correspond bijectively to ternary trees with a given number of middle- and right-edges, and an explicit bijection is provided for this fact. Finally, the in- and outdegree distribution of a single vertex is considered and explicit counting formulas are provided again.

A combinatorial model for exceptional sequences in type A

Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya’s work) to classify exceptional sequences of representations of $Q$, the linearly ordered quiver with $n$ vertices. We also show how to use variations of this model to classify $c$-matrices of $Q$, to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of $c$-matrices, we also give an interpretation of $c$-matrix mutation in terms of our noncrossing trees with directed edges. Les suites exceptionnelles sont certaines suites ordonnées de représentations de carquois. Nous utilisons des arbres aux arêtes étiquetés et aux sommets dans le bord d’un disque (expansion sur le travail de T. Araya) pour classifier les suites exceptionnelles de représentations du carquois linéairement ordonné à $n$ sommets. Nous exploitons des variations de ce modèle pour classifier les $c$-matrices dudit carquois, pour interpréter les suites exceptionnelles comme des extensions linéaires, et pour donner une bijection élémentaire entre les suites exceptionnelles et certaines chaînes dans le réseau des partitions sans croisement. Dans le cas des $c$-matrices, nous donnons également une interprétation de la mutation des $c$-matrices en termes des arbres sans croisement aux arêtes orientés.

Noncrossing Monochromatic Subtrees and Staircases in 0-1 Matrices

The following question is asked by the senior author (Gyárfás (2011)). What is the order of the largest monochromatic noncrossing subtree (caterpillar) that exists in every 2-coloring of the edges of a simple geometric Kn,n? We solve one particular problem asked by Gyárfás (2011): separate the Ramsey number of noncrossing trees from the Ramsey number of noncrossing double stars. We also reformulate the question as a Ramsey-type problem for 0-1 matrices and pose the following conjecture. Every n×n 0-1 matrix contains n−1 zeros or n−1 ones, forming a staircase: a sequence which goes right in rows and down in columns, possibly skipping elements, but not at turning points. We prove this conjecture in some special cases and put forward some related problems as well.

A Decomposition Algorithm for Noncrossing Trees

Based on the classic bijective algorithm for trees due to Chen, we present a decomposition algorithm for noncrossing trees. This leads to a combinatorial interpretation of a formula on noncrossing trees of size $n$ with $k$ descents. We also derive the formula for noncrossing trees of size $n$ with $k$ descents and $i$ leaves, which is a refinement of the formula given by Flajolet and Noy. As an application of our algorithm, we answer a question proposed by Hough, which asks for a bijection between two classes of noncrossing trees with a given number of descents.

An Operational Calculus for the Mould Operad

The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture of the first author about the inverse image of non-crossing trees in the dendriform operad. Finally, we explain a connection with the formalism of noncommutative symmetric functions.