Combinatorial and Probabilistic Approaches to Planar Tanglegrams, Colorful Permutations, and Cosine Functions

Abstract

Through combinatorial and probabilistic approaches, we study the structure of three objects: tanglegrams, colored permutations, and cosine functions. Our work on tanglegrams includes a characterization of planar tanglegram layouts and a method for sampling planar tanglegrams. The former generalizes a result of Lozano, Pinter, Rokhlenko, Valiente, and Ziv-Ukelson for finding a planar layout of a planar tanglegram, and the latter is a planar analog of an algorithm of Billey, Konvalinka, and Matsen for generating tanglegrams uniformly at random. Our work on colored permutations involves analyzing the moments of statistics on conjugacy classes without "short" cycles, with particular emphasis on the descent, major index, and flag-major index statistics. These generalize results of Fulman involving the descent and major index statistics on the symmetric group, as well as certain cases of recent work by Hamaker and Rhoades applying representation theory to the study of moments on conjugacy classes. Our work on cosine functions involves studying minimal cases for the correlation of their signs when the input is scaled by various integers. This is motivated by the study of Schrodinger operators due to Goncalves, Oliveira e Silva, and Steinerberger, who characterized the minimal cases for n=2 cosine functions. We provide the corresponding characterization for n=3 cosine functions.