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%I #16 Jan 10 2024 16:30:25
%S 1,1,3,9,29,97,336,1188,4275,15579,57358,212908,795657,2990221,
%T 11291665,42814783,162920417,621885767,2380348729
%N Number of distinct sorted degree sequences among all n-vertex half-loop-graphs without isolated vertices.
%C In the covering case, these degree sequences, sorted in decreasing order, are the same thing as half-loop-graphical partitions (A321729). An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex.
%C The following are equivalent characteristics for any positive integer n:
%C (1) the prime indices of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops or edges;
%C (2) n can be factored into distinct primes or squarefree semiprimes;
%C (3) the prime signature of n is half-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>
%F a(n) = A029889(n) - A029889(n-1) for n > 0. - _Andrew Howroyd_, Jan 10 2024
%e The a(0) = 1 through a(3) = 9 sorted degree sequences:
%e () (1) (1,1) (1,1,1)
%e (2,1) (2,1,1)
%e (2,2) (2,2,1)
%e (2,2,2)
%e (3,1,1)
%e (3,2,1)
%e (3,2,2)
%e (3,3,2)
%e (3,3,3)
%e For example, the half-loop-graphs
%e {{1},{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{1,2},{1,3}}
%e both have degrees y = (3,2,2), so y is counted under a(3).
%t Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n]&]]],{n,0,5}]
%Y See link for additional cross references.
%Y The version for simple graphs is A004251, covering: A095268.
%Y The non-covering version (it allows isolated vertices) is A029889.
%Y The same partitions counted by sum are conjectured to be A321729.
%Y These graphs are counted by A006125 shifted left, covering: A322661.
%Y The version for full loops is A339844, covering: A339845.
%Y These graphs are ranked by A340018 and A340019.
%Y A006125 counts labeled simple graphs, covering: A006129.
%Y A027187 counts partitions of even length, ranked by A028260.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
%Y A339659 counts graphical partitions of 2n into k parts.
%Y Cf. A062740, A096373, A167171, A320461, A320893, A320921, A320923, A338915, A339560, A339841, A339842.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Dec 27 2020
%E a(7)-a(18) added (using A029889) by _Andrew Howroyd_, Jan 10 2024
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