An Experimental Mathematics Approach to Several Combinatorial Problems

TL;DR

This paper demonstrates how experimental mathematics, using programming and symbolic computation, can investigate and discover properties of various combinatorial problems including parking functions, spanning trees, Quicksort, and peaceable queens.

Contribution

It introduces an experimental mathematics methodology applied to multiple combinatorial problems, showcasing new insights and approaches in the field.

Findings

Analysis of parking functions and their statistics

Application of experimental methods to spanning trees and matrices

Study of Quicksort's running time and peaceable queens problem

Abstract

Experimental mathematics is an experimental approach to mathematics in which programming and symbolic computation are used to investigate mathematical objects, identify properties and patterns, discover facts and formulas and even automatically prove theorems. With an experimental mathematics approach, this dissertation deals with several combinatorial problems and demonstrates the methodology of experimental mathematics. We start with parking functions and their moments of certain statistics. Then we discuss about spanning trees and "almost diagonal" matrices to illustrate the methodology of experimental mathematics. We also apply experimental mathematics to Quicksort algorithms to study the running time. Finally we talk about the interesting peaceable queens problem.