Related papers: Parking functions for trees and mappings
We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the…
Recently, harmonic functions and frequently universal harmonic functions on a tree $T$ have been studied, taking values on a separable Fr\'{e}chet space $E$ over the field $\mathbb{C}$ or $\mathbb{R}$. In the present paper, we allow the…
We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…
Let $\phi$ be a function that maps any non-empty subset $A$ of $\mathbb{R}^2$ to a non-empty subset $\phi(A)$ of $\mathbb{R}^2$. A $\phi$-cover of a set $T=\{T_1, T_2, \dots, T_m\}$ of pairwise non-crossing trees in the plane is a set of…
In 1966, Cummins introduced the "tree graph": the tree graph $\mathbf{T}(G)$ of a graph $G$ (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge,…
Naples parking functions were introduced as a generalization of classical parking functions, in which cars are allowed to park backwards, by checking up to a fixed number of previous spots, before proceeding forward as usual. In this work…
Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called `chronological tree'. In this work, we are interested in the continuum…
In this article, we establish new results on the probabilistic parking model (introduced by Durm\'ic, Han, Harris, Ribeiro, and Yin) with $m$ cars and $n$ parking spots and probability parameter $p\in[0,1]$. For any $ m \leq n$ and $p \in…
Automated parking is a self-driving feature that has been in cars for several years. Parking assistants in currently sold cars fail to park in more complex real-world scenarios and require the driver to move the car to an expected starting…
In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm (FMT*). The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional…
Let $G$ be a connected graph with vertex set $\{0,1,2,...,n\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we…
Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has…
Parking plays a central role in transport policies and has wide-ranging consequences: While the average time spent searching for parking exceeds dozens of hours per driver every year in many Western cities, the associated cruising traffic…
We introduce a novel interpretable tree based algorithm for prediction in a regression setting. Our motivation is to estimate the unknown regression function from a functional decomposition perspective in which the functional components…
A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between $G$-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph $G$. A tree growing…
In a previous paper we considered a sequence of maps on a complete metric space $(X,d)$ and derived an extension of the Banach fixed point theorem. We showed that backward trajectories of maps $X\to X$ converge under mild conditions and…
We propose a tree-based algorithm for classification and regression problems in the context of functional data analysis, which allows to leverage representation learning and multiple splitting rules at the node level, reducing…
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…