Journal of Integer Sequences, Vol. 11 (2008), Article 08.4.6

Enumeration of Unigraphical Partitions

Michael D. Hirschhorn
School of Mathematics
Sydney 2052

James A. Sellers
Department of Mathematics
The Pennsylvania State University
University Park, PA 16802


In the early 1960s, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d1, d2, ..., dn 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 duni(2 m), the number of "unigraphical partitions" with degree sum 2m.

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(Concerned with sequences A000005 A000041 A001399 and A083039 .)

Received September 6 2008; revised version received October 8 2008. Published in Journal of Integer Sequences, October 18 2008.

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