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V. Roitner:
"Studies on several parameters in lattice paths";
Supervisor, Reviewer: B. Gittenberger, C. Banderier, A. Panholzer; Institut für Diskrete Mathematik und Geometrie, 2020; oral examination: 2020-11-19.

The thesis"Studies on several parameters in lattice paths" by Valerie Roitner deals withenumerative as well as asymptotic aspects of directed lattice paths. Several parameters appearingin lattice paths will be analyzed, e.g. the area enclosed by or the number of contacts between twopaths or the number of occurrences of certain patterns in a path.The first chapter gives an overview over the history of lattice path theory as well as anoverview of the applications of lattice paths in mathematical models arising in natural sciences orcomputer science. We will also give a precise definition of lattices and lattice paths. In the secondchapter the methods used in enumerative and asymptotic combinatorics will be introduced:combinatorial classes and their generating functions for exact enumeration as well as singularityanalysis for asymptotic results.There are two lattice path configurations this thesis is particularly focused on: non-intersectingpairs (or tuples) of paths and paths which avoid patterns, i.e., fixed sequences of consecutivesteps. Chapter three deals with non-intersecting pairs of lattice paths. We will derive resultsabout their average number of contacts as well as the average area between them.Chapter four deals with pattern avoidance in lattice paths. First, the vectorial kernel methoddeveloped by Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger willbe introduced, since it is a very powerful tool for enumerating lattice paths avoiding a fixedpattern as well as enumerating the occurrences of a fixed pattern in a lattice path. Then it will begeneralized in two directions: for enumerating lattice paths with longer steps and for enumeratinglattice paths which avoid several patterns at once. The tools developed in this section have alsobeen used to prove a conjecture by David Callan about the asymptotic behavior of the expectednumber of ascents in Schr ̈oder paths.In chapter five we will combine the methods from chapter three and four for studying patternavoidance as well as the lower height in pairs of non-intersecting lattice paths.Some of the results in this thesis have already been published in scientific papers.