Enumeration of Generalised Directed Lattice Paths in a Strip

Abstract

This thesis is about the enumeration of two models of directed lattice paths in a strip. The first problem considered is of path diagrams formed by Dyck paths and columns underneath it, counted with respect to the length of the paths and the sum of the heights of the columns. The enumeration of these path diagrams is related to q-deformed tangent and secant numbers. Generating functions of height-restricted path diagrams are given by convergents of continued fractions. We derive expressions for these convergents in terms of basic hypergeometric functions, leading to a hierarchy of novel identities for basic hypergeometric functions. From these expressions, we also find novel expressions for the infinite continued fractions, leading to a different proof of known enumeration formulas for q-tangent and q-secant numbers. The second problem considered is the enumeration of directed weighted paths in a strip with arbitrary step heights. Here, we find an appealing formula for their generating function in terms of a ratio of two (skew-) Schur functions, evaluated at the roots of the so-called kernel of a linear functional equation. The partitions indexing these Schur functions only depend on the size of the largest up and down steps, and the weights of the individual steps enter via the kernel roots. To aid computation, we express the skew Schur function in this formula in terms of a sum of Schur functions, and give several examples. We also consider an extension where contacts at the boundary are weighted.