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%I #13 Jan 10 2024 16:30:20
%S 1,2,6,16,51,162,554,1918,6843,24688,90342,333308,1239725
%N Number of distinct sorted degree sequences among all n-vertex loop-graphs.
%C In the covering case, these degree sequences, sorted in decreasing order, are the same thing as loop-graphical partitions (A339656). An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices.
%C The following are equivalent characteristics for any positive integer n:
%C (1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;
%C (2) n can be factored into distinct semiprimes;
%C (3) the prime signature of n is loop-graphical.
%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>
%e The a(0) = 1 through a(3) = 16 sorted degree sequences:
%e () (0) (0,0) (0,0,0)
%e (2) (0,2) (0,0,2)
%e (1,1) (0,1,1)
%e (1,3) (0,1,3)
%e (2,2) (0,2,2)
%e (3,3) (0,3,3)
%e (1,1,2)
%e (1,1,4)
%e (1,2,3)
%e (1,3,4)
%e (2,2,2)
%e (2,2,4)
%e (2,3,3)
%e (2,4,4)
%e (3,3,4)
%e (4,4,4)
%e For example, the loop-graphs
%e {{1,1},{2,2},{3,3},{1,2}}
%e {{1,1},{2,2},{3,3},{1,3}}
%e {{1,1},{2,2},{3,3},{2,3}}
%e {{1,1},{2,2},{1,3},{2,3}}
%e {{1,1},{3,3},{1,2},{2,3}}
%e {{2,2},{3,3},{1,2},{1,3}}
%e all have degrees y = (3,3,2), so y is counted under a(3).
%t Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{1,2}]/.{x_Integer}:>{x,x}]]],{n,0,5}]
%Y See link for additional cross references.
%Y The version without loops is A004251, with covering case A095268.
%Y The half-loop version is A029889, with covering case A339843.
%Y Loop-graphs are counted by A322661 and ranked by A320461 and A340020.
%Y The covering case (no zeros) is A339845.
%Y A007717 counts unlabeled multiset partitions into pairs.
%Y A027187 counts partitions of even length, with Heinz numbers A028260.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A101048 counts partitions into semiprimes.
%Y A339655 counts non-loop-graphical partitions of 2n.
%Y A339656 counts loop-graphical partitions of 2n.
%Y A339659 counts graphical partitions of 2n into k parts.
%Y Cf. A001358, A006125, A006129, A062740, A338898, A339841.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Dec 27 2020
%E a(7)-a(12) from _Andrew Howroyd_, Jan 10 2024
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