The numbers of distinct graphical partitions corresponding to graphs on , 2, ... nodes are 0, 1, 2, 7, 20, 71, 240, 871, 3148, ...
(OEIS A095268).
A graphical partition of order is one for which the sum of degrees is . A -graphical partition only exists for even.
It is possible for two topologically distinct graphs to have the same degree
sequence, an example of which is illustrated above.
The numbers
of graphical partitions with total degree , 4, 6, ... are 1, 2, 5, 9, 17, 31, 54, 90, 151, 244, ...
(OEIS A000569).
Writing
for the fraction of partitions of that are graphical, where is the partition function
P, Erdős and Richmond (1989) showed that, as tends to infinity through even
integers,
for which there exists a graph 
having degree
sequence 
.
The number of graphical partitions of length 
is the number of distinct positive degree
sequences, listed in nonincreasing order, realizable by 
-vertex simple
graphs with no isolated vertices.
, 2, ... nodes are 0, 1, 2, 7, 20, 71, 240, 871, 3148, ...
(OEIS A095268).
is one for which the sum of degrees is 
. A 
-graphical partition only exists for even 
.
of graphical partitions with total degree 
, 4, 6, ... are 1, 2, 5, 9, 17, 31, 54, 90, 151, 244, ...
(OEIS A000569).
for the fraction of partitions of 
that are graphical, where 
is the partition function
P, Erdős and Richmond (1989) showed that, as 
tends to infinity through even
integers,
