Lattice Path Combinatorics and Applications

Recent methods used in lattice path combinatorics and various related branches of enumerative combinatorics are grouped together and presented in this volume, together with relevant applications. Contributions are mainly comprised of research articles though an attractive bonus is the inclusion of several captivating, expository articles on the life and mathematical work of leading researchers in lattice path combinatorics and beyond. "The 8th International Conference on Lattice Path Combinatorics and Applications" held in the summer of 2015, provided opportunities for new collaborations and this volume also presents some stimulating ideas from newcomers to the conference series. This book will be of interest to researchers in lattice path combinatorics and enumerative combinatorics including subsets of researchers in mathematics, statistics, operations research, and computer science. The applications of the material covered in this Work extend beyond the primaryaudience to scholars interested in queueing theory, graph theory, rook theory, tilings, partitions, distributions, Markov processes, and more.

Résumé

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 Fq-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 8th 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.

Contenu
Preface.- Lajos Takács: A Tribute (A.M. Haghighi, S.G. Mohanty).- The Distribution of the Local Time of Brownian Motion with Drift (L. Takács).- Reflections on Shreeram Abhyankar (K. Alladi).- My Association and Collaboration with George Andrews (K. Alladi).- A Refinement of the AlladiSchur Theorem (G.E. Andrews).- Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method (C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, M. Wallner).- The Kernel Method for Lattice Paths below a Line of Rational Slope (C. Banderier, M. Wallner).- Enumeration of Colored Dyck Paths via Partial Bell Polynomials (D. Birmajer, J.B. Gil, P.R.W. McNamara, M.D. Weiner).- A Review of the Basic Discrete q-Distributions (Ch.A. Charalambides).- Families of Parking Functions Counted by the Schröder and Baxter Numbers (R. Cori, E. Duchi, S. Rinaldi, V. Guerrini).- Some Tilings, Colorings and Lattice Paths via Stern Polynomials (K. Dilcher, L. Ericksen).- p-Rook Numbers and Cycle Counting in Cp|Sn (J. Haglund, J. Remmel, M. Yoo).- Asymptotic Behaviour of Certain q-Poisson, q-Binomial and Negative q-Binomial Distributions (A. Kyriakoussis, M. Vamvakari).- Asymptotic Estimates for Queueing Systems with Time-Varying Periodic Transition Rates (B. Margolius).- A Combinatorial Analysis of the M/M[m]/1 Queue (G. Mercankosk, G.M. Nair).- Laws Relating Runs, Long Runs, and Steps in Gambler's Ruin, with Persistence in Two Strata (G.J. Morrow).- Paired Patterns in Lattice Paths (R.Pan, J.B. Remmel).