%I #14 Aug 01 2019 00:29:22
%S 1,0,1,1,2,3,5,7,10,15,21,28
%N Number of integer partitions of n that are the vertex-degrees of some set system with no singletons.
%C A set system is a finite set of finite nonempty sets.
%e The a(2) = 1 through a(9) = 15 partitions:
%e (11) (111) (211) (221) (222) (322) (2222) (333)
%e (1111) (2111) (2211) (2221) (3221) (3222)
%e (11111) (3111) (3211) (3311) (3321)
%e (21111) (22111) (22211) (4221)
%e (111111) (31111) (32111) (22221)
%e (211111) (41111) (32211)
%e (1111111) (221111) (33111)
%e (311111) (42111)
%e (2111111) (222111)
%e (11111111) (321111)
%e (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%e (111111111)
%e The a(8) = 10 integer partitions together with a realizing set system for each (the parts of the partition count the appearances of each vertex in the set system):
%e (41111): {{1,2},{1,3},{1,4},{1,5}}
%e (3311): {{1,2},{1,2,3},{1,2,4}}
%e (3221): {{1,2},{1,3},{1,2,3,4}}
%e (32111): {{1,2},{1,3},{1,2,4,5}}
%e (311111): {{1,2},{1,3},{1,4,5,6}}
%e (2222): {{1,2},{3,4},{1,2,3,4}}
%e (22211): {{1,2,3},{1,2,3,4,5}}
%e (221111): {{1,2},{1,2,3,4,5,6}}
%e (2111111): {{1,2},{1,3,4,5,6,7}}
%e (11111111): {{1,2,3,4,5,6,7,8}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t Table[Length[Select[strnorm[n],hyp[#]!={}&]],{n,8}]
%Y Cf. A000070, A000569, A147878, A209816, A283877, A306005, A318361, A320922, A320923, A320924, A321177.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Oct 29 2018