1 - Path Counting—Simple Boundaries

Abstract

This chapter discusses the basic methods of counting lattice paths, namely, the reflection principle, penetrating analysis, recurrence, generating function, the inclusion-exclusion principle, and vector representation. While some of these are standard techniques in solving counting problems, others reveal the usual combinatorial ingenuity that transforms a seemingly difficult problem into one easily amenable to explicit solution. Only enumeration of paths restricted by straight line boundaries are discussed. The classical ballot problem, which notwithstanding its unrealistic formulation as a ballot problem has historically formed the core for path counting problems and has served as a model for various applied problems. The relation of paths with Young tableaux and Fibonacci numbers is reviewed. Also, paths with simple diagonal steps are enumerated, especially with the aid of the “balls into cells” technique. The duality concept is introduced by which a result can be written in a different way when either of the two operators, that is, reversion and conjugation (reflection), or both are applied to the paths under consideration.