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 $$a_1, a_2, \ldots , a_j$$ a 1 , a 2 , … , a j 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 $$i \in \{1,2,3,4\}$$ i ∈ { 1 , 2 , 3 , 4 } , we find generating functions in which the coefficient of $$x^n y^j$$ x n y j is the number of vertices of degree i in all c-graphs with n vertices and j parts.