Related papers: Increasing Forests and Quadrangulations via a Bije…
A bijection $\Phi$ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the…
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…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of…
A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated…
Tree-like tableaux are combinatorial objects that appear in a combinatorial understanding of the PASEP model from statistical mechanics. In this understanding, the corners of the Southeast border correspond to the locations where a particle…
A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters of any prefix is non-negative, together with a permutation \sigma of {1,2,...,n} such…
Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the…
We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes…
The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
It is always a hot and difficult point to improve the accuracy of convolutional neural network model and speed up its convergence. Based on the idea of small world network, a random edge adding algorithm is proposed to improve the…
A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model…
We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected…
A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…
Inspired by the Stochastic processes described by the Feller Coupling and Chinese Restaurant Processes, we create four different bijections from words in the set $[1]\times [2] \times\cdot \times[n]$ to $S_n$. We then compose these maps…
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more…
Random Forests (RF) is a popular machine learning method for classification and regression problems. It involves a bagging application to decision tree models. One of the primary advantages of the Random Forests model is the reduction in…
We study the local limit of the fixed-point forest, a tree structure associated to a simple sorting algorithm on permutations. This local limit can be viewed as an infinite random tree that can be constructed from a Poisson point process…
The isolation forest algorithm for outlier detection exploits a simple yet effective observation: if taking some multivariate data and making uniformly random cuts across the feature space recursively, it will take fewer such random cuts…
Traditional Quartet Puzzling algorithms use maximum likelihood methods to reconstruct quartet trees, and a puzzling algorithm to combine these quartets into a tree for the full collection of $n$ taxa. We propose a variation of Quartet…