Ballot theorem and lattice path crossings

1978 ◽  
Vol 6 (1) ◽  
pp. 87-90
Author(s):  
Malcolm Kern ◽  
Stanley Walter
Keyword(s):  
Lattice Path ◽  
Ballot Theorem

Modelling the order of scoring in team sports

2020 ◽  
Author(s):  
Kengo Hamada ◽  
Ken-ichi Tanaka
Keyword(s):  
Team Sports ◽  
Poisson Model ◽  
Ice Hockey ◽  
Lattice Path ◽  
Independence Assumption ◽  
Ballot Theorem ◽  
Lattice Path Enumeration ◽  
Sports Data

Abstract This paper considers sports matches in which two teams compete to score more points within a set amount of time (e.g. football, ice hockey). We focus on the order in which the competing teams score during the match (order of scoring). This type of order of scoring problem has not been addressed previously, and doing so here gives new insights into sports matches. For example, our analysis can deal with a situation that spectators find matches that involve comebacks particularly exciting. To describe such problems mathematically, we formulate the probabilities of (i) the favourite team leading throughout the match and (ii) the favourite team falling behind the opposing team but then making a comeback. These probabilities are derived using an independent Poisson model and lattice path enumeration, the latter of which involves the well-known ballot theorem. The independence assumption allows lattice path enumeration to be applied directly to the Poisson model and various scoring patterns to be addressed. We confirm that the values obtained from the proposed models agree well with actual sports data from football, futsal and ice hockey.

scholarly journals Lattice Path Proofs of Extended Bressoud-Wei and Koike Skew Schur Function Identities

10.37236/534 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
A. M. Hamel ◽  
R. C. King
Keyword(s):  
Lattice Path ◽  
Schur Function ◽  
Skew Schur Function

A recent paper of the present authors provides extensions to two classical determinantal results of Bressoud and Wei, and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs.

scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

Ballots, queues and random graphs

1989 ◽  
Vol 26 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Real Number ◽  
Random Graph ◽  
Random Graphs ◽  
Limit Distribution ◽  
Positive Real Number ◽  
Positive Real ◽  
Ballot Theorem

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n(s) the number of vertices in the union of all those components of Γn(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n(s) and the limit distribution of ρ n(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

Lengths of runs and waiting time distributions by using Pólya-Eggenberger sampling scheme

2002 ◽  
Vol 39 (3-4) ◽  
pp. 309-332 ◽  
Author(s):  
K. Sen ◽  
Manju L. Agarwal ◽  
S. Chakraborty
Keyword(s):  
Waiting Time ◽  
Lattice Path ◽  
Sampling Scheme ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Waiting Time Distributions ◽  
Joint Distributions ◽  
First Occurrence

In this paper, joint distributions of number of success runs of length k and number of failure runs of length k' are obtained by using combinatorial techniques including lattice path approach under Pólya-Eggenberger model. Some of its particular cases, for different values of the parameters, are derived. Sooner and later waiting time problems and joint distributions of number of success runs of various types until first occurrence of consecutive success runs of specified length under the model are obtained. The sooner and later waiting time problems for Bernoulli trials (see Ebneshahrashoob and Sobel [3]) and joint distributions of the type discussed by Uchiada and Aki [11] are shown as particular cases. Assuming Ln and Sn to be the lengths of longest and smallest success runs, respectively, in a sample of size n drawn by Pólya-Eggenberger sampling scheme, the joint distributions of Ln and  Sn as well as distribution of M=max(Ln,Fn)n, where Fn is the length of longest failure run, are also  obtained.

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