Enumeration of the degree sequences of non-separable graphs and connected graphs

In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d₁, d₂, ..., d_n to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions d_{n s} (2 m) and d_c (2 m), the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: d_{n s} (2 m) = p (2 m) - p (2 m - 1) - underover(∑, j = 0, m - 2) p (j) and d_c (2 m) = p (2 m) - p (m - 1) - 2 underover(∑, j = 0, m - 2) p (j) where p (j) is the number of unrestricted integer partitions of j.