A solution to the tennis ball problem

We present a complete solution to the so-called tennis ball problem, which is equivalent to counting lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions. Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem.Comment: 9 pages, Late

A solution to the tennis ball problem

We present a complete solution to the so-called tennis ball problem, which is equivalent to counting the number of lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions. Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem.Postprint (published version

On sphere-filling ropes

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.Comment: 15 pages, 8 figure

The Tennis Ball Problem

AbstractMallows and Shapiro, (J. Integer Sequences2 (1999)) have recently considered what they dubbed the problem of balls on the lawn. Our object is to explore a natural generalization, the s-tennis ball problem, which reduces to that considered by Mallows and Shapiro in the case s=2. We show how this generalization is connected with s-ary trees, and employ the notion of generating trees to obtain a solution expressed in terms of generating functions

Set-Valued Tableaux & Generalized Catalan Numbers

Standard set-valued Young tableaux are a generalization of standard Young tableaux in which cells may contain more than one integer, with the added conditions that every integer at position $(i,j)$ must be smaller than every integer at positions $(i,j+1)$ and $(i+1,j)$. This paper explores the combinatorics of standard set-valued Young tableaux with two-rows, and how those tableaux may be used to provide new combinatorial interpretations of generalized Catalan numbers. New combinatorial interpretations are provided for the two-parameter Fuss-Catalan numbers (Raney numbers), the rational Catalan numbers, and the solution to the so-called "generalized tennis ball problem". Methodologies are then introduced for the enumeration of standard set-valued Young tableaux, prompting explicit formulas for the general two-row case. The paper closes by drawing a bijection between arbitrary classes of two-row standard set-valued Young tableaux and collections of two-dimensional lattice paths that lie weakly below a unique maximal path

Automatic annotation of tennis games: An integration of audio, vision, and learning

Fully automatic annotation of tennis game using broadcast video is a task with a great potential but with enormous challenges. In this paper we describe our approach to this task, which integrates computer vision, machine listening, and machine learning. At the low level processing, we improve upon our previously proposed state-of-the-art tennis ball tracking algorithm and employ audio signal processing techniques to detect key events and construct features for classifying the events. At high level analysis, we model event classification as a sequence labelling problem, and investigate four machine learning techniques using simulated event sequences. Finally, we evaluate our proposed approach on three real world tennis games, and discuss the interplay between audio, vision and learning. To the best of our knowledge, our system is the only one that can annotate tennis game at such a detailed level

Why do dogs (Canis familiaris) select the empty container in an observational learning task?

Many argue that dogs show unique susceptibility to human communicative signals that make them suitable for being engaged in complex co-operation with humans. It has also been revealed that socially provided information is particularly effective in influencing the behaviour of dogs even when the human’s action demonstration conveys inefficient or mistaken solution of task. It is unclear, however, how the communicative nature of the demonstration context and the presence of the human demonstrator affect the dogs’ object-choice behaviour in observational learning situations. In order to unfold the effects of these factors, 76 adult pet dogs could observe a communicative or a non-communicative demonstration in which the human retrieved a tennis ball from under an opaque container while manipulating another distant and obviously empty (transparent) one. Subjects were then allowed to choose either in the presence of the demonstrator or after she left the room. Results showed a significant main effect of the demonstration context (presence or absence of the human’s communicative signals), and we also found some evidence for the response-modifying effect of the presence of the human demonstrator during the dogs’ choice. That is, dogs predominantly chose the baited container, but if the demonstration context was communicative and the human was present during the dogs’ choice, subjects’ tendency to select the baited container has been reduced. In agreement with the studies showing sensitivity to human’s communicative signals in dogs, these findings point to a special form of social influence in observational learning situations when it comes to learning about causally opaque and less efficient (compared to what comes natural to the dog) action demonstrations