cp's OEIS Frontend

A029889 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).

%I A029889 #25 Apr 18 2021 17:25:34
%S A029889 1,2,5,14,43,140,476,1664,5939,21518,78876,291784,1087441,4077662,
%T A029889 15369327,58184110,221104527,842990294,3223339023
%N A029889 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 A029889 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 A029889 R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
%H A029889 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 A029889 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>
%H A029889 Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%H A029889 <a href="/index/Gra#graph_part">Index entries for sequences related to graphical partitions</a>
%F A029889 Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.
%F A029889 a(n) = A029890(n) + A029891(n). - _Andrew Howroyd_, Apr 18 2021
%e A029889 From _Gus Wiseman_, Dec 31 2020: (Start)
%e A029889 The a(0) = 1 through a(3) = 14 sorted degree sequences:
%e A029889   ()  (0)  (0,0)  (0,0,0)
%e A029889       (1)  (1,0)  (1,0,0)
%e A029889            (1,1)  (1,1,0)
%e A029889            (2,1)  (2,1,0)
%e A029889            (2,2)  (2,2,0)
%e A029889                   (1,1,1)
%e A029889                   (2,1,1)
%e A029889                   (3,1,1)
%e A029889                   (2,2,1)
%e A029889                   (3,2,1)
%e A029889                   (2,2,2)
%e A029889                   (3,2,2)
%e A029889                   (3,3,2)
%e A029889                   (3,3,3)
%e A029889 For example, the half-loop-graph
%e A029889   {{1,3},{3}}
%e A029889 has degrees (1,0,2), so (2,1,0) is counted under a(3). The half-loop-graphs
%e A029889   {{1},{1,2},{1,3},{2,3}}
%e A029889   {{1},{2},{3},{1,2},{1,3}}
%e A029889 both have degrees (3,2,2), so (3,2,2) is counted under a(3).
%e A029889 (End)
%t A029889 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 A029889 Cf. A000569, A004250, A029890, A029891.
%Y A029889 Non-half-loop-graphical partitions are conjectured to be counted by A321728.
%Y A029889 The covering case (no zeros) is A339843.
%Y A029889 MM-numbers of half-loop-graphs are given by A340018 and A340019.
%Y A029889 A004251 counts degree sequences of graphs, with covering case A095268.
%Y A029889 A320663 counts unlabeled multiset partitions into singletons/pairs.
%Y A029889 A339659 is a triangle counting graphical partitions.
%Y A029889 A339844 counts degree sequences of loop-graphs, with covering case A339845.
%Y A029889 Cf. A006125, A006129, A027187, A028260, A062740, A096373, A322661, A339560.
%K A029889 nonn,more
%O A029889 0,2
%A A029889 torsten.sillke(AT)lhsystems.com
%E A029889 a(0) = 1 prepended by _Gus Wiseman_, Dec 31 2020