Geodesic
Enumeration of unigraphical partitions
Journal of integer sequences, Tome 11 (2008) no. 4
In the early 1960s, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers $d_{1}, d_{2}, \dots , d_{n}$ 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 $2m$.
Classification : 05A15, 05A17, 05C75, 11P81
Keywords: partition, degree sequence, graph, unigraphical, generating function (Concerned with sequences and ) 
@article{JIS_2008__11_4_a4,
     author = {Hirschhorn,  Michael D. and Sellers,  James A.},
     title = {Enumeration of unigraphical partitions},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {4},
     zbl = {1148.05008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_4_a4/}
}
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Hirschhorn,  Michael D.; Sellers,  James A. Enumeration of unigraphical partitions. Journal of integer sequences, Tome 11 (2008) no. 4. http://geodesic.mathdoc.fr/item/JIS_2008__11_4_a4/