

A029889


Number of graphical partitions (degreevectors for graphs with n vertices, allowing selfloops which count as degree 1; or possible ordered rowsum vectors for a symmetric 01 matrix).


17



1, 2, 5, 14, 43, 140, 476, 1664, 5939, 21518, 78876, 291784, 1087441, 4077662, 15369327, 58184110, 221104527, 842990294, 3223339023
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OFFSET

0,2


COMMENTS

I call loops of degree one halfloops, so these are halfloopgraphs or graphs with halfloops.  Gus Wiseman, Dec 31 2020


REFERENCES

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.


LINKS



FORMULA

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of BrualdiRyser.


EXAMPLE

The a(0) = 1 through a(3) = 14 sorted degree sequences:
() (0) (0,0) (0,0,0)
(1) (1,0) (1,0,0)
(1,1) (1,1,0)
(2,1) (2,1,0)
(2,2) (2,2,0)
(1,1,1)
(2,1,1)
(3,1,1)
(2,2,1)
(3,2,1)
(2,2,2)
(3,2,2)
(3,3,2)
(3,3,3)
For example, the halfloopgraph
{{1,3},{3}}
has degrees (1,0,2), so (2,1,0) is counted under a(3). The halfloopgraphs
{{1},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3}}
both have degrees (3,2,2), so (3,2,2) is counted under a(3).
(End)


MATHEMATICA

Table[Length[Union[Sort[Table[Count[Join@@#, i], {i, n}]]&/@Subsets[Subsets[Range[n], {1, 2}]]]], {n, 0, 5}] (* Gus Wiseman, Dec 31 2020 *)


CROSSREFS

Nonhalfloopgraphical partitions are conjectured to be counted by A321728.
The covering case (no zeros) is A339843.
A004251 counts degree sequences of graphs, with covering case A095268.
A320663 counts unlabeled multiset partitions into singletons/pairs.
A339659 is a triangle counting graphical partitions.
A339844 counts degree sequences of loopgraphs, with covering case A339845.


KEYWORD

nonn,more


AUTHOR

torsten.sillke(AT)lhsystems.com


EXTENSIONS



STATUS

approved



