Several Topics in Experimental Mathematics

TL;DR

This thesis applies experimental mathematics to diverse problems including random trees, lattice paths, summation algorithms, and a graph conjecture, providing new computational methods and partial proofs.

Contribution

It introduces automated procedures for moments of random trees, finds recurrences for lattice path counts, extends summation algorithms, and proves a case of the bunk bed conjecture.

Findings

Automated computation of moments for Galton-Watson trees.

Recurrences for lattice paths with rational slopes.

Partial proof of the bunk bed conjecture for specific cases.

Abstract

This thesis deals with applications of experimental mathematics to a number of problems. The first problem is related to random graph statistics. We consider a certain class of Galton-Watson random trees and look at the total height statistic. We provide an automated procedure for computing values of the moments of this statistic. Next, we investigate several problems related to lattice paths staying below a line of rational slope. These results are largely data-based. Using the generated data, we are able to find recurrences for the number of such paths for the cases of slopes 3/2 and 5/2. There is also investigation of a generalization of these problems to three dimensions. We also examine generalizations of Sister Celine's method and Gosper's algorithm for evaluating summations. For both, we greatly extend the classes of applicable functions and applications to proving, or reproving in an automated way, interesting combinatorial problems. For the generalization of Sister Celine's method, we allow summations of arbitrary products of hypergeometric terms and linear recurrent sequences with rational coefficients. We also show a partial result related to the bunk bed conjecture, a problem concerning random finite graphs. Let $G$ be a finite graph. Remove edges from $G\square K_2$ independently and with the same probability. In $G\square K_2$, there is an edge placed between all vertices of $G$ and the corresponding vertex in a copy of $G$. Then, label these vertices as either $(v,0)$ or $(v,1)$ for each $v\in V(G)$. The conjecture says that for any $x,y \in V(G)$, it is least as likely to have $(x,0)$ connected to $(y,0)$ as to have $(x,0)$ connected to $(y,1)$. We prove the conjecture in the case that only two of the edges going between the two copes of $G$ are retained.