OFFSET
0,2
COMMENTS
I call loops of degree one half-loops, so these are half-loop-graphs or graphs with half-loops. - Gus Wiseman, Dec 31 2020
REFERENCES
R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
LINKS
T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
Eric Weisstein's World of Mathematics, Degree Sequence.
FORMULA
Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.
EXAMPLE
From Gus Wiseman, Dec 31 2020: (Start)
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 half-loop-graph
{{1,3},{3}}
has degrees (1,0,2), so (2,1,0) is counted under a(3). The half-loop-graphs
{{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
KEYWORD
nonn,more
AUTHOR
torsten.sillke(AT)lhsystems.com
EXTENSIONS
a(0) = 1 prepended by Gus Wiseman, Dec 31 2020
STATUS
approved