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A339845 Number of distinct sorted degree sequences among all n-vertex loop-graphs without isolated vertices.

%I A339845 #13 Jan 10 2024 16:30:15
%S A339845 1,1,4,10,35,111,392,1364,4925,17845,65654,242966,906417
%N A339845 Number of distinct sorted degree sequences among all n-vertex loop-graphs without isolated vertices.
%C A339845 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 A339845 The following are equivalent characteristics for any positive integer n:
%C A339845 (1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;
%C A339845 (2) n can be factored into distinct semiprimes;
%C A339845 (3) the prime signature of n is loop-graphical.
%H A339845 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>
%H A339845 Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%F A339845 a(n) = A339844(n) - A339844(n-1) for n > 0. - _Andrew Howroyd_, Jan 10 2024
%e A339845 The a(0) = 1 through a(3) = 10 sorted degree sequences:
%e A339845   ()  (2)  (1,1)  (1,1,2)
%e A339845            (1,3)  (1,1,4)
%e A339845            (2,2)  (1,2,3)
%e A339845            (3,3)  (1,3,4)
%e A339845                   (2,2,2)
%e A339845                   (2,2,4)
%e A339845                   (2,3,3)
%e A339845                   (2,4,4)
%e A339845                   (3,3,4)
%e A339845                   (4,4,4)
%e A339845 For example, the loop-graphs
%e A339845   {{1,1},{2,2},{3,3},{1,2}}
%e A339845   {{1,1},{2,2},{3,3},{1,3}}
%e A339845   {{1,1},{2,2},{3,3},{2,3}}
%e A339845   {{1,1},{2,2},{1,3},{2,3}}
%e A339845   {{1,1},{3,3},{1,2},{2,3}}
%e A339845   {{2,2},{3,3},{1,2},{1,3}}
%e A339845 all have degrees y = (3,3,2), so y is counted under a(3).
%t A339845 Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Select[Subsets[Subsets[Range[n],{1,2}]/.{x_Integer}:>{x,x}],Union@@#==Range[n]&]]],{n,0,5}]
%Y A339845 See link for additional cross references.
%Y A339845 The version without loops is A004251, with covering case A095268.
%Y A339845 The half-loop version is A029889, with covering case A339843.
%Y A339845 Loop-graphs are counted by A322661 and ranked by A320461 and A340020.
%Y A339845 Counting the same partitions by sum gives A339656.
%Y A339845 These partitions are ranked by A339658.
%Y A339845 The non-covering case (zeros allowed) is A339844.
%Y A339845 A007717 counts unlabeled multiset partitions into pairs.
%Y A339845 A027187 counts partitions of even length, ranked by A028260.
%Y A339845 A058696 counts partitions of even numbers, ranked by A300061.
%Y A339845 A101048 counts partitions into semiprimes.
%Y A339845 A339655 counts non-loop-graphical partitions of 2n.
%Y A339845 A339659 counts graphical partitions of 2n into k parts.
%Y A339845 Cf. A001358, A006125, A006129, A062740, A338898, A339841.
%K A339845 nonn,more
%O A339845 0,3
%A A339845 _Gus Wiseman_, Dec 27 2020
%E A339845 a(7)-a(12) from _Andrew Howroyd_, Jan 10 2024