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Enumeration of Unigraphical Partitions
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Michael D. Hirschhorn

School of Mathematics

UNSW

Sydney 2052

Australia

James A. Sellers

Department of Mathematics

The Pennsylvania State University

University Park, PA 16802

USA

**Abstract:**

In the early 1960s, S. L. Hakimi proved necessary and sufficient
conditions for a given sequence of positive integers
*d*_{1}, *d*_{2}, ...,
*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 2*m*.

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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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