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Number of graphical partitions (degree-vectors for graphs with n vertices, allowing self-loops which count as degree 1; or possible ordered row-sum vectors for a symmetric 0-1 matrix).
17

%I #25 Apr 18 2021 17:25:34

%S 1,2,5,14,43,140,476,1664,5939,21518,78876,291784,1087441,4077662,

%T 15369327,58184110,221104527,842990294,3223339023

%N Number of graphical partitions (degree-vectors for graphs with n vertices, allowing self-loops which count as degree 1; or possible ordered row-sum vectors for a symmetric 0-1 matrix).

%C I call loops of degree one half-loops, so these are half-loop-graphs or graphs with half-loops. - _Gus Wiseman_, Dec 31 2020

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

%H T. M. Barnes and C. D. Savage, <a href="https://doi.org/10.37236/1205">A recurrence for counting graphical partitions</a>, Electronic J. Combinatorics, 2 (1995).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>

%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>

%H <a href="/index/Gra#graph_part">Index entries for sequences related to graphical partitions</a>

%F Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

%F a(n) = A029890(n) + A029891(n). - _Andrew Howroyd_, Apr 18 2021

%e From _Gus Wiseman_, Dec 31 2020: (Start)

%e The a(0) = 1 through a(3) = 14 sorted degree sequences:

%e () (0) (0,0) (0,0,0)

%e (1) (1,0) (1,0,0)

%e (1,1) (1,1,0)

%e (2,1) (2,1,0)

%e (2,2) (2,2,0)

%e (1,1,1)

%e (2,1,1)

%e (3,1,1)

%e (2,2,1)

%e (3,2,1)

%e (2,2,2)

%e (3,2,2)

%e (3,3,2)

%e (3,3,3)

%e For example, the half-loop-graph

%e {{1,3},{3}}

%e has degrees (1,0,2), so (2,1,0) is counted under a(3). The half-loop-graphs

%e {{1},{1,2},{1,3},{2,3}}

%e {{1},{2},{3},{1,2},{1,3}}

%e both have degrees (3,2,2), so (3,2,2) is counted under a(3).

%e (End)

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

%Y Cf. A000569, A004250, A029890, A029891.

%Y Non-half-loop-graphical partitions are conjectured to be counted by A321728.

%Y The covering case (no zeros) is A339843.

%Y MM-numbers of half-loop-graphs are given by A340018 and A340019.

%Y A004251 counts degree sequences of graphs, with covering case A095268.

%Y A320663 counts unlabeled multiset partitions into singletons/pairs.

%Y A339659 is a triangle counting graphical partitions.

%Y A339844 counts degree sequences of loop-graphs, with covering case A339845.

%Y Cf. A006125, A006129, A027187, A028260, A062740, A096373, A322661, A339560.

%K nonn,more

%O 0,2

%A torsten.sillke(AT)lhsystems.com

%E a(0) = 1 prepended by _Gus Wiseman_, Dec 31 2020