Lattice Path Combinatorics and Applications

✍ Scribed by George E. Andrews, Christian Krattenthaler, Alan Krinik

• Publisher: Springer International Publishing
• Year: 2019
• Tongue: English
• Leaves: 443
• Series: Developments in Mathematics 58
• Edition: 1st ed.
• Category: Library

✦ Synopsis

The most recent methods in various branches of lattice path and enumerative combinatorics along with relevant applications are nicely grouped together and represented in this research contributed volume. Contributions to this edited volume will be mainly research articles however it will also include several captivating, expository articles (along with pictures) on the life and mathematical work of leading researchers in lattice path combinatorics and beyond. There will be four or five expository articles in memory of Shreeram Shankar Abhyankar and Philippe Flajolet and honoring George Andrews and Lajos Takács. There may be another brief article in memory of Professors Jagdish Narayan Srivastava and Joti Lal Jain.

New research results include the kernel method developed by Flajolet and others for counting different classes of lattice paths continues to produce new results in counting lattice paths. The recent investigation of Fishburn numbers has led to interesting counting interpretations and a family of fascinating congruences. Formulas for new methods to obtain the number of F_q-rational points of Schubert varieties in Grassmannians continues to have research interest and will be presented here. Topics to be included are far reaching and will include lattice path enumeration, tilings, bijections between paths and other combinatoric structures, non-intersecting lattice paths, varieties, Young tableaux, partitions, enumerative combinatorics, discrete distributions, applications to queueing theory and other continuous time models, graph theory and applications. Many leading mathematicians who spoke at the conference from which this volume derives, are expected to send contributions including. This volume also presents the stimulating ideas of some exciting newcomers to the Lattice Path Combinatorics Conference series; “The 8^th Conference on Lattice Path Combinatorics and Applications” provided opportunities for new collaborations; some of the products of these collaborations will also appear in this book.

This book will have interest for researchers in lattice path combinatorics and enumerative combinatorics. This will include subsets of researchers in mathematics, statistics, operations research and computer science. The applications of the material covered in this edited volume extends beyond the primary audience to scholars interested queuing theory, graph theory, tiling, partitions, distributions, etc. An attractive bonus within our book is the collection of special articles describing the top recent researchers in this area of study and documenting the interesting history of who, when and how these beautiful combinatorial results were originally discovered.

✦ Table of Contents

Front Matter ....Pages i-xxv
Professor Lajos Takács: A Tribute (Aliakbar Montazer Haghighi, Sri Gopal Mohanty)....Pages 1-28
The Distribution of the Local Time of Brownian Motion with Drift (Lajos Takács)....Pages 29-42
Reflections on Shreeram Abhyankar (Krishnaswami Alladi)....Pages 43-46
My Association and Collaboration with George Andrews (Krishnaswami Alladi)....Pages 47-70
A Refinement of the Alladi–Schur Theorem (George E. Andrews)....Pages 71-77
Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method (Cyril Banderier, Christian Krattenthaler, Alan Krinik, Dmitry Kruchinin, Vladimir Kruchinin, David Nguyen et al.)....Pages 78-118
The Kernel Method for Lattice Paths Below a Line of Rational Slope (Cyril Banderier, Michael Wallner)....Pages 119-154
Enumeration of Colored Dyck Paths Via Partial Bell Polynomials (Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara, Michael D. Weiner)....Pages 155-165
A Review of the Basic Discrete q-Distributions (Ch. A. Charalambides)....Pages 166-193
Families of Parking Functions Counted by the Schröder and Baxter Numbers (Robert Cori, Enrica Duchi, Veronica Guerrini, Simone Rinaldi)....Pages 194-225
Some Tilings, Colorings and Lattice Paths via Stern Polynomials (Karl Dilcher, Larry Ericksen)....Pages 226-249
p-Rook Numbers and Cycle Counting in (C_p \wr S_n) (James Haglund, Jeffrey B. Remmel, Meesue Yoo)....Pages 250-282
Asymptotic Behaviour of Certain q-Poisson, q-Binomial and Negative q-Binomial Distributions (Andreas Kyriakoussis, Malvina Vamvakari)....Pages 283-306
Asymptotic Estimates for Queueing Systems with Time-Varying Periodic Transition Rates (Barbara Margolius)....Pages 307-326
A Combinatorial Analysis of the M/M(^{[m]})/1 Queue (Guven Mercankosk, Gopalan M. Nair)....Pages 327-342
Laws Relating Runs, Long Runs, and Steps in Gambler’s Ruin, with Persistence in Two Strata (Gregory J. Morrow)....Pages 343-381
Paired Patterns in Lattice Paths (Ran Pan, Jeffrey B. Remmel)....Pages 382-418

✦ Subjects

Mathematics; Combinatorics; Number Theory; Graph Theory; Convex and Discrete Geometry; Probability Theory and Stochastic Processes