In the early 1960s, 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 unique graph (that is, one and only one graph realization exists for such a degree sequence). Our goal in this note is to utilize Hakimi's characterization to prove a closed formula for the function d_uni(2 m), the number of "unigraphical partitions" with degree sum 2m.