Andrew Lohr, Rutgers University

Abstract

First, we'll talk about the total height statistic on a certain family of random graphs. We are able to get Maple to compute moments of this statistic. Taking limits, we confirm, via elementary methods, the fact, due to David Aldous, that the limiting (scaled) distributions are all the same. Second, we'll talk about generalizations of Sister Celine's method and Gosper's algorithm for evaluating summations. For both, we greatly extend the classes of applicable functions. For the generalization of Sister Celine's method, we allow summations of arbitrary products of hypergeometric terms and linear recurrent sequences with rational coefficients. For the extension of Gosper's algorithm, we extend it from solely hypergeometric sequences to any multi-basic sequence. For both, we give numerous applications to proving, or reproving in an automated way, interesting combinatorial problems. We'll also discuss the bunk bed conjecture, showing that in some special cases that were previously unknown, the statement of the conjecture holds.