A Recurrence for Counting Graphical Partitions

Abstract

In this paper, we give a recurrence to enumerate the set $G(n)$ of partitions of a positive even integer $n$ which are the degree sequences of simple graphs. The recurrence gives rise to an algorithm to compute the number of elements of $G(n)$ in time $O(n^4)$ using space $O(n^3)$. This appears to be the first method for computing $|G(n)|$ in time bounded by a polynomial in $n$, and it has enabled us to tabulate $|G(n)|$ for even $n \leq 220$.