The degree of vertices in graphs of integer partitions

Abstract

This paper unifies disparate areas of combinatorics. Namely, graph theory and integer partition theory. A partition of an integer n is a finite sequence a1,a2,…,aj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1, a_2, \ldots , a_j$$\end{document} of weakly increasing positive integers (called parts) whose sum is n. In order to achieve a unification, a specific type of graph called a c-graph is defined for each partition. The main thrust of the paper is investigating the degrees of vertices in c-graphs. For each of the possible fixed degrees i∈{1,2,3,4}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \in \{1,2,3,4\}$$\end{document}, we find generating functions in which the coefficient of xnyj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^n y^j$$\end{document} is the number of vertices of degree i in all c-graphs with n vertices and j parts.